Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  symgfixf1 Structured version   Visualization version   GIF version

Theorem symgfixf1 17680
 Description: The mapping of a permutation of a set fixing an element to a permutation of the set without the fixed element is a 1-1 function. (Contributed by AV, 4-Jan-2019.)
Hypotheses
Ref Expression
symgfixf.p 𝑃 = (Base‘(SymGrp‘𝑁))
symgfixf.q 𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}
symgfixf.s 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))
symgfixf.h 𝐻 = (𝑞𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))
Assertion
Ref Expression
symgfixf1 (𝐾𝑁𝐻:𝑄1-1𝑆)
Distinct variable groups:   𝐾,𝑞   𝑃,𝑞   𝑁,𝑞   𝑄,𝑞   𝑆,𝑞
Allowed substitution hint:   𝐻(𝑞)

Proof of Theorem symgfixf1
Dummy variables 𝑔 𝑝 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 symgfixf.p . . 3 𝑃 = (Base‘(SymGrp‘𝑁))
2 symgfixf.q . . 3 𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}
3 symgfixf.s . . 3 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))
4 symgfixf.h . . 3 𝐻 = (𝑞𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))
51, 2, 3, 4symgfixf 17679 . 2 (𝐾𝑁𝐻:𝑄𝑆)
64fvtresfn 6193 . . . . . 6 (𝑔𝑄 → (𝐻𝑔) = (𝑔 ↾ (𝑁 ∖ {𝐾})))
74fvtresfn 6193 . . . . . 6 (𝑝𝑄 → (𝐻𝑝) = (𝑝 ↾ (𝑁 ∖ {𝐾})))
86, 7eqeqan12d 2626 . . . . 5 ((𝑔𝑄𝑝𝑄) → ((𝐻𝑔) = (𝐻𝑝) ↔ (𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾}))))
98adantl 481 . . . 4 ((𝐾𝑁 ∧ (𝑔𝑄𝑝𝑄)) → ((𝐻𝑔) = (𝐻𝑝) ↔ (𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾}))))
10 vex 3176 . . . . . . 7 𝑔 ∈ V
111, 2symgfixelq 17676 . . . . . . 7 (𝑔 ∈ V → (𝑔𝑄 ↔ (𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾)))
1210, 11ax-mp 5 . . . . . 6 (𝑔𝑄 ↔ (𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾))
13 vex 3176 . . . . . . 7 𝑝 ∈ V
141, 2symgfixelq 17676 . . . . . . 7 (𝑝 ∈ V → (𝑝𝑄 ↔ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)))
1513, 14ax-mp 5 . . . . . 6 (𝑝𝑄 ↔ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))
1612, 15anbi12i 729 . . . . 5 ((𝑔𝑄𝑝𝑄) ↔ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)))
17 f1ofn 6051 . . . . . . . . . . 11 (𝑔:𝑁1-1-onto𝑁𝑔 Fn 𝑁)
1817adantr 480 . . . . . . . . . 10 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → 𝑔 Fn 𝑁)
19 f1ofn 6051 . . . . . . . . . . 11 (𝑝:𝑁1-1-onto𝑁𝑝 Fn 𝑁)
2019adantr 480 . . . . . . . . . 10 ((𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾) → 𝑝 Fn 𝑁)
2118, 20anim12i 588 . . . . . . . . 9 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (𝑔 Fn 𝑁𝑝 Fn 𝑁))
22 difss 3699 . . . . . . . . 9 (𝑁 ∖ {𝐾}) ⊆ 𝑁
2321, 22jctir 559 . . . . . . . 8 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → ((𝑔 Fn 𝑁𝑝 Fn 𝑁) ∧ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
2423adantl 481 . . . . . . 7 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → ((𝑔 Fn 𝑁𝑝 Fn 𝑁) ∧ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
25 fvreseq 6227 . . . . . . 7 (((𝑔 Fn 𝑁𝑝 Fn 𝑁) ∧ (𝑁 ∖ {𝐾}) ⊆ 𝑁) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) ↔ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)))
2624, 25syl 17 . . . . . 6 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) ↔ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)))
27 f1of 6050 . . . . . . . . . . . 12 (𝑔:𝑁1-1-onto𝑁𝑔:𝑁𝑁)
2827adantr 480 . . . . . . . . . . 11 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → 𝑔:𝑁𝑁)
29 f1of 6050 . . . . . . . . . . . 12 (𝑝:𝑁1-1-onto𝑁𝑝:𝑁𝑁)
3029adantr 480 . . . . . . . . . . 11 ((𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾) → 𝑝:𝑁𝑁)
31 fdm 5964 . . . . . . . . . . . 12 (𝑔:𝑁𝑁 → dom 𝑔 = 𝑁)
32 fdm 5964 . . . . . . . . . . . 12 (𝑝:𝑁𝑁 → dom 𝑝 = 𝑁)
3331, 32anim12i 588 . . . . . . . . . . 11 ((𝑔:𝑁𝑁𝑝:𝑁𝑁) → (dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁))
3428, 30, 33syl2an 493 . . . . . . . . . 10 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁))
35 eqtr3 2631 . . . . . . . . . 10 ((dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁) → dom 𝑔 = dom 𝑝)
3634, 35syl 17 . . . . . . . . 9 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → dom 𝑔 = dom 𝑝)
3736ad2antlr 759 . . . . . . . 8 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → dom 𝑔 = dom 𝑝)
38 simpr 476 . . . . . . . . . 10 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖))
39 eqtr3 2631 . . . . . . . . . . . 12 (((𝑔𝐾) = 𝐾 ∧ (𝑝𝐾) = 𝐾) → (𝑔𝐾) = (𝑝𝐾))
4039ad2ant2l 778 . . . . . . . . . . 11 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (𝑔𝐾) = (𝑝𝐾))
4140ad2antlr 759 . . . . . . . . . 10 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → (𝑔𝐾) = (𝑝𝐾))
42 fveq2 6103 . . . . . . . . . . . . . 14 (𝑖 = 𝐾 → (𝑔𝑖) = (𝑔𝐾))
43 fveq2 6103 . . . . . . . . . . . . . 14 (𝑖 = 𝐾 → (𝑝𝑖) = (𝑝𝐾))
4442, 43eqeq12d 2625 . . . . . . . . . . . . 13 (𝑖 = 𝐾 → ((𝑔𝑖) = (𝑝𝑖) ↔ (𝑔𝐾) = (𝑝𝐾)))
4544ralunsn 4360 . . . . . . . . . . . 12 (𝐾𝑁 → (∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔𝑖) = (𝑝𝑖) ↔ (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖) ∧ (𝑔𝐾) = (𝑝𝐾))))
4645adantr 480 . . . . . . . . . . 11 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → (∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔𝑖) = (𝑝𝑖) ↔ (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖) ∧ (𝑔𝐾) = (𝑝𝐾))))
4746adantr 480 . . . . . . . . . 10 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → (∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔𝑖) = (𝑝𝑖) ↔ (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖) ∧ (𝑔𝐾) = (𝑝𝐾))))
4838, 41, 47mpbir2and 959 . . . . . . . . 9 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → ∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔𝑖) = (𝑝𝑖))
49 f1odm 6054 . . . . . . . . . . . . . 14 (𝑔:𝑁1-1-onto𝑁 → dom 𝑔 = 𝑁)
5049adantr 480 . . . . . . . . . . . . 13 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → dom 𝑔 = 𝑁)
5150adantr 480 . . . . . . . . . . . 12 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → dom 𝑔 = 𝑁)
52 difsnid 4282 . . . . . . . . . . . . 13 (𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁)
5352eqcomd 2616 . . . . . . . . . . . 12 (𝐾𝑁𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
5451, 53sylan9eqr 2666 . . . . . . . . . . 11 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → dom 𝑔 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
5554adantr 480 . . . . . . . . . 10 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → dom 𝑔 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
5655raleqdv 3121 . . . . . . . . 9 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → (∀𝑖 ∈ dom 𝑔(𝑔𝑖) = (𝑝𝑖) ↔ ∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔𝑖) = (𝑝𝑖)))
5748, 56mpbird 246 . . . . . . . 8 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → ∀𝑖 ∈ dom 𝑔(𝑔𝑖) = (𝑝𝑖))
58 f1ofun 6052 . . . . . . . . . . . 12 (𝑔:𝑁1-1-onto𝑁 → Fun 𝑔)
5958adantr 480 . . . . . . . . . . 11 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → Fun 𝑔)
60 f1ofun 6052 . . . . . . . . . . . 12 (𝑝:𝑁1-1-onto𝑁 → Fun 𝑝)
6160adantr 480 . . . . . . . . . . 11 ((𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾) → Fun 𝑝)
6259, 61anim12i 588 . . . . . . . . . 10 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (Fun 𝑔 ∧ Fun 𝑝))
6362ad2antlr 759 . . . . . . . . 9 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → (Fun 𝑔 ∧ Fun 𝑝))
64 eqfunfv 6224 . . . . . . . . 9 ((Fun 𝑔 ∧ Fun 𝑝) → (𝑔 = 𝑝 ↔ (dom 𝑔 = dom 𝑝 ∧ ∀𝑖 ∈ dom 𝑔(𝑔𝑖) = (𝑝𝑖))))
6563, 64syl 17 . . . . . . . 8 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → (𝑔 = 𝑝 ↔ (dom 𝑔 = dom 𝑝 ∧ ∀𝑖 ∈ dom 𝑔(𝑔𝑖) = (𝑝𝑖))))
6637, 57, 65mpbir2and 959 . . . . . . 7 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → 𝑔 = 𝑝)
6766ex 449 . . . . . 6 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖) → 𝑔 = 𝑝))
6826, 67sylbid 229 . . . . 5 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) → 𝑔 = 𝑝))
6916, 68sylan2b 491 . . . 4 ((𝐾𝑁 ∧ (𝑔𝑄𝑝𝑄)) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) → 𝑔 = 𝑝))
709, 69sylbid 229 . . 3 ((𝐾𝑁 ∧ (𝑔𝑄𝑝𝑄)) → ((𝐻𝑔) = (𝐻𝑝) → 𝑔 = 𝑝))
7170ralrimivva 2954 . 2 (𝐾𝑁 → ∀𝑔𝑄𝑝𝑄 ((𝐻𝑔) = (𝐻𝑝) → 𝑔 = 𝑝))
72 dff13 6416 . 2 (𝐻:𝑄1-1𝑆 ↔ (𝐻:𝑄𝑆 ∧ ∀𝑔𝑄𝑝𝑄 ((𝐻𝑔) = (𝐻𝑝) → 𝑔 = 𝑝)))
735, 71, 72sylanbrc 695 1 (𝐾𝑁𝐻:𝑄1-1𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  {csn 4125   ↦ cmpt 4643  dom cdm 5038   ↾ cres 5040  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  Basecbs 15695  SymGrpcsymg 17620 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-tset 15787  df-symg 17621 This theorem is referenced by:  symgfixf1o  17683
 Copyright terms: Public domain W3C validator