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Theorem f1eqcocnv 6510
Description: Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
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 6123 . . . 4 (𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
2 coeq2 5240 . . . . 5 (𝐹 = 𝐺 → (𝐹𝐹) = (𝐹𝐺))
32eqeq1d 2623 . . . 4 (𝐹 = 𝐺 → ((𝐹𝐹) = ( I ↾ 𝐴) ↔ (𝐹𝐺) = ( I ↾ 𝐴)))
41, 3syl5ibcom 235 . . 3 (𝐹:𝐴1-1𝐵 → (𝐹 = 𝐺 → (𝐹𝐺) = ( I ↾ 𝐴)))
54adantr 481 . 2 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → (𝐹 = 𝐺 → (𝐹𝐺) = ( I ↾ 𝐴)))
6 f1fn 6059 . . . . . . 7 (𝐺:𝐴1-1𝐵𝐺 Fn 𝐴)
76adantl 482 . . . . . 6 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → 𝐺 Fn 𝐴)
87adantr 481 . . . . 5 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐴)) → 𝐺 Fn 𝐴)
9 f1fn 6059 . . . . . . 7 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
109adantr 481 . . . . . 6 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → 𝐹 Fn 𝐴)
1110adantr 481 . . . . 5 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐴)) → 𝐹 Fn 𝐴)
12 equid 1936 . . . . . . . . . 10 𝑥 = 𝑥
13 resieq 5366 . . . . . . . . . 10 ((𝑥𝐴𝑥𝐴) → (𝑥( I ↾ 𝐴)𝑥𝑥 = 𝑥))
1412, 13mpbiri 248 . . . . . . . . 9 ((𝑥𝐴𝑥𝐴) → 𝑥( I ↾ 𝐴)𝑥)
1514anidms 676 . . . . . . . 8 (𝑥𝐴𝑥( I ↾ 𝐴)𝑥)
1615adantl 482 . . . . . . 7 ((((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐴)) ∧ 𝑥𝐴) → 𝑥( I ↾ 𝐴)𝑥)
17 breq 4615 . . . . . . . 8 ((𝐹𝐺) = ( I ↾ 𝐴) → (𝑥(𝐹𝐺)𝑥𝑥( I ↾ 𝐴)𝑥))
1817ad2antlr 762 . . . . . . 7 ((((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐴)) ∧ 𝑥𝐴) → (𝑥(𝐹𝐺)𝑥𝑥( I ↾ 𝐴)𝑥))
1916, 18mpbird 247 . . . . . 6 ((((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐴)) ∧ 𝑥𝐴) → 𝑥(𝐹𝐺)𝑥)
20 fnfun 5946 . . . . . . . . . . . . . . . 16 (𝐺 Fn 𝐴 → Fun 𝐺)
217, 20syl 17 . . . . . . . . . . . . . . 15 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → Fun 𝐺)
2221adantr 481 . . . . . . . . . . . . . 14 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → Fun 𝐺)
23 fndm 5948 . . . . . . . . . . . . . . . . 17 (𝐺 Fn 𝐴 → dom 𝐺 = 𝐴)
247, 23syl 17 . . . . . . . . . . . . . . . 16 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → dom 𝐺 = 𝐴)
2524eleq2d 2684 . . . . . . . . . . . . . . 15 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → (𝑥 ∈ dom 𝐺𝑥𝐴))
2625biimpar 502 . . . . . . . . . . . . . 14 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → 𝑥 ∈ dom 𝐺)
27 funopfvb 6196 . . . . . . . . . . . . . 14 ((Fun 𝐺𝑥 ∈ dom 𝐺) → ((𝐺𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
2822, 26, 27syl2anc 692 . . . . . . . . . . . . 13 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → ((𝐺𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
2928bicomd 213 . . . . . . . . . . . 12 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 ↔ (𝐺𝑥) = 𝑦))
30 df-br 4614 . . . . . . . . . . . 12 (𝑥𝐺𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐺)
31 eqcom 2628 . . . . . . . . . . . 12 (𝑦 = (𝐺𝑥) ↔ (𝐺𝑥) = 𝑦)
3229, 30, 313bitr4g 303 . . . . . . . . . . 11 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → (𝑥𝐺𝑦𝑦 = (𝐺𝑥)))
3332biimpd 219 . . . . . . . . . 10 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → (𝑥𝐺𝑦𝑦 = (𝐺𝑥)))
34 fnfun 5946 . . . . . . . . . . . . . . . 16 (𝐹 Fn 𝐴 → Fun 𝐹)
3510, 34syl 17 . . . . . . . . . . . . . . 15 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → Fun 𝐹)
3635adantr 481 . . . . . . . . . . . . . 14 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → Fun 𝐹)
37 fndm 5948 . . . . . . . . . . . . . . . . 17 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3810, 37syl 17 . . . . . . . . . . . . . . . 16 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → dom 𝐹 = 𝐴)
3938eleq2d 2684 . . . . . . . . . . . . . . 15 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → (𝑥 ∈ dom 𝐹𝑥𝐴))
4039biimpar 502 . . . . . . . . . . . . . 14 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → 𝑥 ∈ dom 𝐹)
41 funopfvb 6196 . . . . . . . . . . . . . 14 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐹𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
4236, 40, 41syl2anc 692 . . . . . . . . . . . . 13 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → ((𝐹𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
43 df-br 4614 . . . . . . . . . . . . 13 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
4442, 43syl6rbbr 279 . . . . . . . . . . . 12 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → (𝑥𝐹𝑦 ↔ (𝐹𝑥) = 𝑦))
45 vex 3189 . . . . . . . . . . . . 13 𝑦 ∈ V
46 vex 3189 . . . . . . . . . . . . 13 𝑥 ∈ V
4745, 46brcnv 5265 . . . . . . . . . . . 12 (𝑦𝐹𝑥𝑥𝐹𝑦)
48 eqcom 2628 . . . . . . . . . . . 12 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
4944, 47, 483bitr4g 303 . . . . . . . . . . 11 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → (𝑦𝐹𝑥𝑦 = (𝐹𝑥)))
5049biimpd 219 . . . . . . . . . 10 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → (𝑦𝐹𝑥𝑦 = (𝐹𝑥)))
5133, 50anim12d 585 . . . . . . . . 9 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → ((𝑥𝐺𝑦𝑦𝐹𝑥) → (𝑦 = (𝐺𝑥) ∧ 𝑦 = (𝐹𝑥))))
5251eximdv 1843 . . . . . . . 8 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → (∃𝑦(𝑥𝐺𝑦𝑦𝐹𝑥) → ∃𝑦(𝑦 = (𝐺𝑥) ∧ 𝑦 = (𝐹𝑥))))
5346, 46brco 5252 . . . . . . . 8 (𝑥(𝐹𝐺)𝑥 ↔ ∃𝑦(𝑥𝐺𝑦𝑦𝐹𝑥))
54 fvex 6158 . . . . . . . . 9 (𝐺𝑥) ∈ V
5554eqvinc 3313 . . . . . . . 8 ((𝐺𝑥) = (𝐹𝑥) ↔ ∃𝑦(𝑦 = (𝐺𝑥) ∧ 𝑦 = (𝐹𝑥)))
5652, 53, 553imtr4g 285 . . . . . . 7 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ 𝑥𝐴) → (𝑥(𝐹𝐺)𝑥 → (𝐺𝑥) = (𝐹𝑥)))
5756adantlr 750 . . . . . 6 ((((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐴)) ∧ 𝑥𝐴) → (𝑥(𝐹𝐺)𝑥 → (𝐺𝑥) = (𝐹𝑥)))
5819, 57mpd 15 . . . . 5 ((((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐴)) ∧ 𝑥𝐴) → (𝐺𝑥) = (𝐹𝑥))
598, 11, 58eqfnfvd 6270 . . . 4 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐴)) → 𝐺 = 𝐹)
6059eqcomd 2627 . . 3 (((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐴)) → 𝐹 = 𝐺)
6160ex 450 . 2 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → ((𝐹𝐺) = ( I ↾ 𝐴) → 𝐹 = 𝐺))
625, 61impbid 202 1 ((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → (𝐹 = 𝐺 ↔ (𝐹𝐺) = ( I ↾ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  cop 4154   class class class wbr 4613   I cid 4984  ccnv 5073  dom cdm 5074  cres 5076  ccom 5078  Fun wfun 5841   Fn wfn 5842  1-1wf1 5844  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855
This theorem is referenced by:  weisoeq  6559
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