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Theorem tfrlem5 7340
Description: Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.)
Hypothesis
Ref Expression
tfrlem.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Assertion
Ref Expression
tfrlem5 ((𝑔𝐴𝐴) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
Distinct variable groups:   𝑓,𝑔,𝑥,𝑦,,𝑢,𝑣,𝐹   𝐴,𝑔,
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑣,𝑢,𝑓)

Proof of Theorem tfrlem5
Dummy variables 𝑧 𝑎 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . 3 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
2 vex 3175 . . 3 𝑔 ∈ V
31, 2tfrlem3a 7337 . 2 (𝑔𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))))
4 vex 3175 . . 3 ∈ V
51, 4tfrlem3a 7337 . 2 (𝐴 ↔ ∃𝑤 ∈ On ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎))))
6 reeanv 3085 . . 3 (∃𝑧 ∈ On ∃𝑤 ∈ On ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ↔ (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ∃𝑤 ∈ On ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))))
7 simp2ll 1120 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑔 Fn 𝑧)
8 simp3l 1081 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥𝑔𝑢)
9 fnbr 5893 . . . . . . . . 9 ((𝑔 Fn 𝑧𝑥𝑔𝑢) → 𝑥𝑧)
107, 8, 9syl2anc 690 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥𝑧)
11 simp2rl 1122 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → Fn 𝑤)
12 simp3r 1082 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥𝑣)
13 fnbr 5893 . . . . . . . . 9 (( Fn 𝑤𝑥𝑣) → 𝑥𝑤)
1411, 12, 13syl2anc 690 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥𝑤)
1510, 14elind 3759 . . . . . . 7 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥 ∈ (𝑧𝑤))
16 onin 5657 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤) ∈ On)
17163ad2ant1 1074 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑧𝑤) ∈ On)
18 fnfun 5888 . . . . . . . . . 10 (𝑔 Fn 𝑧 → Fun 𝑔)
197, 18syl 17 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → Fun 𝑔)
20 inss1 3794 . . . . . . . . . 10 (𝑧𝑤) ⊆ 𝑧
21 fndm 5890 . . . . . . . . . . 11 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
227, 21syl 17 . . . . . . . . . 10 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → dom 𝑔 = 𝑧)
2320, 22syl5sseqr 3616 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑧𝑤) ⊆ dom 𝑔)
2419, 23jca 552 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (Fun 𝑔 ∧ (𝑧𝑤) ⊆ dom 𝑔))
25 fnfun 5888 . . . . . . . . . 10 ( Fn 𝑤 → Fun )
2611, 25syl 17 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → Fun )
27 inss2 3795 . . . . . . . . . 10 (𝑧𝑤) ⊆ 𝑤
28 fndm 5890 . . . . . . . . . . 11 ( Fn 𝑤 → dom = 𝑤)
2911, 28syl 17 . . . . . . . . . 10 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → dom = 𝑤)
3027, 29syl5sseqr 3616 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑧𝑤) ⊆ dom )
3126, 30jca 552 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (Fun ∧ (𝑧𝑤) ⊆ dom ))
32 simp2lr 1121 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎)))
33 ssralv 3628 . . . . . . . . 9 ((𝑧𝑤) ⊆ 𝑧 → (∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎)) → ∀𝑎 ∈ (𝑧𝑤)(𝑔𝑎) = (𝐹‘(𝑔𝑎))))
3420, 32, 33mpsyl 65 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎 ∈ (𝑧𝑤)(𝑔𝑎) = (𝐹‘(𝑔𝑎)))
35 simp2rr 1123 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))
36 ssralv 3628 . . . . . . . . 9 ((𝑧𝑤) ⊆ 𝑤 → (∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)) → ∀𝑎 ∈ (𝑧𝑤)(𝑎) = (𝐹‘(𝑎))))
3727, 35, 36mpsyl 65 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎 ∈ (𝑧𝑤)(𝑎) = (𝐹‘(𝑎)))
3817, 24, 31, 34, 37tfrlem1 7336 . . . . . . 7 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎 ∈ (𝑧𝑤)(𝑔𝑎) = (𝑎))
39 fveq2 6088 . . . . . . . . 9 (𝑎 = 𝑥 → (𝑔𝑎) = (𝑔𝑥))
40 fveq2 6088 . . . . . . . . 9 (𝑎 = 𝑥 → (𝑎) = (𝑥))
4139, 40eqeq12d 2624 . . . . . . . 8 (𝑎 = 𝑥 → ((𝑔𝑎) = (𝑎) ↔ (𝑔𝑥) = (𝑥)))
4241rspcv 3277 . . . . . . 7 (𝑥 ∈ (𝑧𝑤) → (∀𝑎 ∈ (𝑧𝑤)(𝑔𝑎) = (𝑎) → (𝑔𝑥) = (𝑥)))
4315, 38, 42sylc 62 . . . . . 6 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑔𝑥) = (𝑥))
44 funbrfv 6129 . . . . . . 7 (Fun 𝑔 → (𝑥𝑔𝑢 → (𝑔𝑥) = 𝑢))
4519, 8, 44sylc 62 . . . . . 6 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑔𝑥) = 𝑢)
46 funbrfv 6129 . . . . . . 7 (Fun → (𝑥𝑣 → (𝑥) = 𝑣))
4726, 12, 46sylc 62 . . . . . 6 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑥) = 𝑣)
4843, 45, 473eqtr3d 2651 . . . . 5 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑢 = 𝑣)
49483exp 1255 . . . 4 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣)))
5049rexlimivv 3017 . . 3 (∃𝑧 ∈ On ∃𝑤 ∈ On ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
516, 50sylbir 223 . 2 ((∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ∃𝑤 ∈ On ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
523, 5, 51syl2anb 494 1 ((𝑔𝐴𝐴) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  {cab 2595  wral 2895  wrex 2896  cin 3538  wss 3539   class class class wbr 4577  dom cdm 5028  cres 5030  Oncon0 5626  Fun wfun 5784   Fn wfn 5785  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-iota 5754  df-fun 5792  df-fn 5793  df-fv 5798
This theorem is referenced by:  tfrlem7  7343
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