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Theorem f1eqcocnv 7321
Description: Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Proof shortened by Wolf Lammen, 29-May-2024.)
Assertion
Ref Expression
f1eqcocnv ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → (𝐹 = 𝐺 ↔ (𝐹𝐺) = ( I ↾ 𝐴)))

Proof of Theorem f1eqcocnv
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1cocnv1 6878 . . . 4 (𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
2 coeq2 5869 . . . . 5 (𝐹 = 𝐺 → (𝐹𝐹) = (𝐹𝐺))
32eqeq1d 2739 . . . 4 (𝐹 = 𝐺 → ((𝐹𝐹) = ( I ↾ 𝐴) ↔ (𝐹𝐺) = ( I ↾ 𝐴)))
41, 3syl5ibcom 245 . . 3 (𝐹:𝐴1-1𝐵 → (𝐹 = 𝐺 → (𝐹𝐺) = ( I ↾ 𝐴)))
54adantr 480 . 2 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → (𝐹 = 𝐺 → (𝐹𝐺) = ( I ↾ 𝐴)))
6 f1fn 6805 . . . . . . 7 (𝐺:𝐴1-1𝐵𝐺 Fn 𝐴)
76adantl 481 . . . . . 6 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → 𝐺 Fn 𝐴)
87adantr 480 . . . . 5 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐴)) → 𝐺 Fn 𝐴)
9 f1fn 6805 . . . . . . 7 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
109adantr 480 . . . . . 6 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → 𝐹 Fn 𝐴)
1110adantr 480 . . . . 5 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐴)) → 𝐹 Fn 𝐴)
12 equid 2011 . . . . . . . . . 10 𝑥 = 𝑥
13 resieq 6008 . . . . . . . . . 10 ((𝑥𝐴𝑥𝐴) → (𝑥( I ↾ 𝐴)𝑥𝑥 = 𝑥))
1412, 13mpbiri 258 . . . . . . . . 9 ((𝑥𝐴𝑥𝐴) → 𝑥( I ↾ 𝐴)𝑥)
1514anidms 566 . . . . . . . 8 (𝑥𝐴𝑥( I ↾ 𝐴)𝑥)
1615adantl 481 . . . . . . 7 ((((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐴)) ∧ 𝑥𝐴) → 𝑥( I ↾ 𝐴)𝑥)
17 breq 5145 . . . . . . . 8 ((𝐹𝐺) = ( I ↾ 𝐴) → (𝑥(𝐹𝐺)𝑥𝑥( I ↾ 𝐴)𝑥))
1817ad2antlr 727 . . . . . . 7 ((((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐴)) ∧ 𝑥𝐴) → (𝑥(𝐹𝐺)𝑥𝑥( I ↾ 𝐴)𝑥))
1916, 18mpbird 257 . . . . . 6 ((((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐴)) ∧ 𝑥𝐴) → 𝑥(𝐹𝐺)𝑥)
20 fnfun 6668 . . . . . . . . . . . . . . 15 (𝐺 Fn 𝐴 → Fun 𝐺)
217, 20syl 17 . . . . . . . . . . . . . 14 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → Fun 𝐺)
227fndmd 6673 . . . . . . . . . . . . . . . 16 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → dom 𝐺 = 𝐴)
2322eleq2d 2827 . . . . . . . . . . . . . . 15 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → (𝑥 ∈ dom 𝐺𝑥𝐴))
2423biimpar 477 . . . . . . . . . . . . . 14 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → 𝑥 ∈ dom 𝐺)
25 funopfvb 6963 . . . . . . . . . . . . . 14 ((Fun 𝐺𝑥 ∈ dom 𝐺) → ((𝐺𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
2621, 24, 25syl2an2r 685 . . . . . . . . . . . . 13 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → ((𝐺𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
2726bicomd 223 . . . . . . . . . . . 12 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 ↔ (𝐺𝑥) = 𝑦))
28 df-br 5144 . . . . . . . . . . . 12 (𝑥𝐺𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)
29 eqcom 2744 . . . . . . . . . . . 12 (𝑦 = (𝐺𝑥) ↔ (𝐺𝑥) = 𝑦)
3027, 28, 293bitr4g 314 . . . . . . . . . . 11 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → (𝑥𝐺𝑦𝑦 = (𝐺𝑥)))
3130biimpd 229 . . . . . . . . . 10 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → (𝑥𝐺𝑦𝑦 = (𝐺𝑥)))
32 df-br 5144 . . . . . . . . . . . . 13 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
33 fnfun 6668 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝐴 → Fun 𝐹)
3410, 33syl 17 . . . . . . . . . . . . . 14 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → Fun 𝐹)
3510fndmd 6673 . . . . . . . . . . . . . . . 16 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → dom 𝐹 = 𝐴)
3635eleq2d 2827 . . . . . . . . . . . . . . 15 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → (𝑥 ∈ dom 𝐹𝑥𝐴))
3736biimpar 477 . . . . . . . . . . . . . 14 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → 𝑥 ∈ dom 𝐹)
38 funopfvb 6963 . . . . . . . . . . . . . 14 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐹𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
3934, 37, 38syl2an2r 685 . . . . . . . . . . . . 13 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → ((𝐹𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
4032, 39bitr4id 290 . . . . . . . . . . . 12 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → (𝑥𝐹𝑦 ↔ (𝐹𝑥) = 𝑦))
41 vex 3484 . . . . . . . . . . . . 13 𝑦 ∈ V
42 vex 3484 . . . . . . . . . . . . 13 𝑥 ∈ V
4341, 42brcnv 5893 . . . . . . . . . . . 12 (𝑦𝐹𝑥𝑥𝐹𝑦)
44 eqcom 2744 . . . . . . . . . . . 12 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
4540, 43, 443bitr4g 314 . . . . . . . . . . 11 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → (𝑦𝐹𝑥𝑦 = (𝐹𝑥)))
4645biimpd 229 . . . . . . . . . 10 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → (𝑦𝐹𝑥𝑦 = (𝐹𝑥)))
4731, 46anim12d 609 . . . . . . . . 9 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → ((𝑥𝐺𝑦𝑦𝐹𝑥) → (𝑦 = (𝐺𝑥) ∧ 𝑦 = (𝐹𝑥))))
4847eximdv 1917 . . . . . . . 8 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → (∃𝑦(𝑥𝐺𝑦𝑦𝐹𝑥) → ∃𝑦(𝑦 = (𝐺𝑥) ∧ 𝑦 = (𝐹𝑥))))
4942, 42brco 5881 . . . . . . . 8 (𝑥(𝐹𝐺)𝑥 ↔ ∃𝑦(𝑥𝐺𝑦𝑦𝐹𝑥))
50 fvex 6919 . . . . . . . . 9 (𝐺𝑥) ∈ V
5150eqvinc 3649 . . . . . . . 8 ((𝐺𝑥) = (𝐹𝑥) ↔ ∃𝑦(𝑦 = (𝐺𝑥) ∧ 𝑦 = (𝐹𝑥)))
5248, 49, 513imtr4g 296 . . . . . . 7 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → (𝑥(𝐹𝐺)𝑥 → (𝐺𝑥) = (𝐹𝑥)))
5352adantlr 715 . . . . . 6 ((((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐴)) ∧ 𝑥𝐴) → (𝑥(𝐹𝐺)𝑥 → (𝐺𝑥) = (𝐹𝑥)))
5419, 53mpd 15 . . . . 5 ((((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐴)) ∧ 𝑥𝐴) → (𝐺𝑥) = (𝐹𝑥))
558, 11, 54eqfnfvd 7054 . . . 4 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐴)) → 𝐺 = 𝐹)
5655eqcomd 2743 . . 3 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐴)) → 𝐹 = 𝐺)
5756ex 412 . 2 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → ((𝐹𝐺) = ( I ↾ 𝐴) → 𝐹 = 𝐺))
585, 57impbid 212 1 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → (𝐹 = 𝐺 ↔ (𝐹𝐺) = ( I ↾ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  cop 4632   class class class wbr 5143   I cid 5577  ccnv 5684  dom cdm 5685  cres 5687  ccom 5689  Fun wfun 6555   Fn wfn 6556  1-1wf1 6558  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by:  weisoeq  7375
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