Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tz7.48-2 Structured version   Visualization version   GIF version

Theorem tz7.48-2 7424
 Description: Proposition 7.48(2) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.) (Revised by David Abernethy, 5-May-2013.)
Hypothesis
Ref Expression
tz7.48.1 𝐹 Fn On
Assertion
Ref Expression
tz7.48-2 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → Fun 𝐹)
Distinct variable group:   𝑥,𝐹
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem tz7.48-2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssid 3587 . . 3 On ⊆ On
2 onelon 5665 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
32ancoms 468 . . . . . . . 8 ((𝑦𝑥𝑥 ∈ On) → 𝑦 ∈ On)
4 tz7.48.1 . . . . . . . . . . 11 𝐹 Fn On
5 fndm 5904 . . . . . . . . . . 11 (𝐹 Fn On → dom 𝐹 = On)
64, 5ax-mp 5 . . . . . . . . . 10 dom 𝐹 = On
76eleq2i 2680 . . . . . . . . 9 (𝑦 ∈ dom 𝐹𝑦 ∈ On)
8 fnfun 5902 . . . . . . . . . . . . 13 (𝐹 Fn On → Fun 𝐹)
94, 8ax-mp 5 . . . . . . . . . . . 12 Fun 𝐹
10 funfvima 6396 . . . . . . . . . . . 12 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)))
119, 10mpan 702 . . . . . . . . . . 11 (𝑦 ∈ dom 𝐹 → (𝑦𝑥 → (𝐹𝑦) ∈ (𝐹𝑥)))
1211impcom 445 . . . . . . . . . 10 ((𝑦𝑥𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ (𝐹𝑥))
13 eleq1a 2683 . . . . . . . . . . 11 ((𝐹𝑦) ∈ (𝐹𝑥) → ((𝐹𝑥) = (𝐹𝑦) → (𝐹𝑥) ∈ (𝐹𝑥)))
14 eldifn 3695 . . . . . . . . . . 11 ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) ∈ (𝐹𝑥))
1513, 14nsyli 154 . . . . . . . . . 10 ((𝐹𝑦) ∈ (𝐹𝑥) → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) = (𝐹𝑦)))
1612, 15syl 17 . . . . . . . . 9 ((𝑦𝑥𝑦 ∈ dom 𝐹) → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) = (𝐹𝑦)))
177, 16sylan2br 492 . . . . . . . 8 ((𝑦𝑥𝑦 ∈ On) → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) = (𝐹𝑦)))
183, 17syldan 486 . . . . . . 7 ((𝑦𝑥𝑥 ∈ On) → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ (𝐹𝑥) = (𝐹𝑦)))
1918expimpd 627 . . . . . 6 (𝑦𝑥 → ((𝑥 ∈ On ∧ (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → ¬ (𝐹𝑥) = (𝐹𝑦)))
2019com12 32 . . . . 5 ((𝑥 ∈ On ∧ (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))
2120ralrimiv 2948 . . . 4 ((𝑥 ∈ On ∧ (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦))
2221ralimiaa 2935 . . 3 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦))
234tz7.48lem 7423 . . 3 ((On ⊆ On ∧ ∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)) → Fun (𝐹 ↾ On))
241, 22, 23sylancr 694 . 2 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → Fun (𝐹 ↾ On))
25 fnrel 5903 . . . . . 6 (𝐹 Fn On → Rel 𝐹)
264, 25ax-mp 5 . . . . 5 Rel 𝐹
276eqimssi 3622 . . . . 5 dom 𝐹 ⊆ On
28 relssres 5357 . . . . 5 ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
2926, 27, 28mp2an 704 . . . 4 (𝐹 ↾ On) = 𝐹
3029cnveqi 5219 . . 3 (𝐹 ↾ On) = 𝐹
3130funeqi 5824 . 2 (Fun (𝐹 ↾ On) ↔ Fun 𝐹)
3224, 31sylib 207 1 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → Fun 𝐹)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∖ cdif 3537   ⊆ wss 3540  ◡ccnv 5037  dom cdm 5038   ↾ cres 5040   “ cima 5041  Rel wrel 5043  Oncon0 5640  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804 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-pr 4833 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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  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-ord 5643  df-on 5644  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fv 5812 This theorem is referenced by:  tz7.48-3  7426
 Copyright terms: Public domain W3C validator