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Theorem foeqcnvco 5806
Description: Condition for function equality in terms of vanishing of the composition with the converse. EDITORIAL: Is there a relation-algebraic proof of this? (Contributed by Stefan O'Rear, 12-Feb-2015.)
Assertion
Ref Expression
foeqcnvco ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → (𝐹 = 𝐺 ↔ (𝐹𝐺) = ( I ↾ 𝐵)))

Proof of Theorem foeqcnvco
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fococnv2 5501 . . . 4 (𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))
2 cnveq 4815 . . . . . 6 (𝐹 = 𝐺𝐹 = 𝐺)
32coeq2d 4803 . . . . 5 (𝐹 = 𝐺 → (𝐹𝐹) = (𝐹𝐺))
43eqeq1d 2197 . . . 4 (𝐹 = 𝐺 → ((𝐹𝐹) = ( I ↾ 𝐵) ↔ (𝐹𝐺) = ( I ↾ 𝐵)))
51, 4syl5ibcom 155 . . 3 (𝐹:𝐴onto𝐵 → (𝐹 = 𝐺 → (𝐹𝐺) = ( I ↾ 𝐵)))
65adantr 276 . 2 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → (𝐹 = 𝐺 → (𝐹𝐺) = ( I ↾ 𝐵)))
7 fofn 5454 . . . . 5 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
87ad2antrr 488 . . . 4 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹 Fn 𝐴)
9 fofn 5454 . . . . 5 (𝐺:𝐴onto𝐵𝐺 Fn 𝐴)
109ad2antlr 489 . . . 4 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐺 Fn 𝐴)
119adantl 277 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐺 Fn 𝐴)
12 fnopfv 5661 . . . . . . . . . . . 12 ((𝐺 Fn 𝐴𝑥𝐴) → ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺)
1311, 12sylan 283 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺)
149anim1i 340 . . . . . . . . . . . . 13 ((𝐺:𝐴onto𝐵𝑥𝐴) → (𝐺 Fn 𝐴𝑥𝐴))
1514adantll 476 . . . . . . . . . . . 12 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐺 Fn 𝐴𝑥𝐴))
16 funfvex 5546 . . . . . . . . . . . . . . 15 ((Fun 𝐺𝑥 ∈ dom 𝐺) → (𝐺𝑥) ∈ V)
1716funfni 5330 . . . . . . . . . . . . . 14 ((𝐺 Fn 𝐴𝑥𝐴) → (𝐺𝑥) ∈ V)
18 vex 2754 . . . . . . . . . . . . . 14 𝑥 ∈ V
19 brcnvg 4822 . . . . . . . . . . . . . 14 (((𝐺𝑥) ∈ V ∧ 𝑥 ∈ V) → ((𝐺𝑥)𝐺𝑥𝑥𝐺(𝐺𝑥)))
2017, 18, 19sylancl 413 . . . . . . . . . . . . 13 ((𝐺 Fn 𝐴𝑥𝐴) → ((𝐺𝑥)𝐺𝑥𝑥𝐺(𝐺𝑥)))
21 df-br 4018 . . . . . . . . . . . . 13 (𝑥𝐺(𝐺𝑥) ↔ ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺)
2220, 21bitrdi 196 . . . . . . . . . . . 12 ((𝐺 Fn 𝐴𝑥𝐴) → ((𝐺𝑥)𝐺𝑥 ↔ ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺))
2315, 22syl 14 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ((𝐺𝑥)𝐺𝑥 ↔ ⟨𝑥, (𝐺𝑥)⟩ ∈ 𝐺))
2413, 23mpbird 167 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐺𝑥)𝐺𝑥)
257adantr 276 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐹 Fn 𝐴)
26 fnopfv 5661 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
2725, 26sylan 283 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
28 df-br 4018 . . . . . . . . . . 11 (𝑥𝐹(𝐹𝑥) ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
2927, 28sylibr 134 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → 𝑥𝐹(𝐹𝑥))
30 breq2 4021 . . . . . . . . . . . 12 (𝑦 = 𝑥 → ((𝐺𝑥)𝐺𝑦 ↔ (𝐺𝑥)𝐺𝑥))
31 breq1 4020 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑦𝐹(𝐹𝑥) ↔ 𝑥𝐹(𝐹𝑥)))
3230, 31anbi12d 473 . . . . . . . . . . 11 (𝑦 = 𝑥 → (((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥)) ↔ ((𝐺𝑥)𝐺𝑥𝑥𝐹(𝐹𝑥))))
3318, 32spcev 2846 . . . . . . . . . 10 (((𝐺𝑥)𝐺𝑥𝑥𝐹(𝐹𝑥)) → ∃𝑦((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥)))
3424, 29, 33syl2anc 411 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ∃𝑦((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥)))
3515, 17syl 14 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐺𝑥) ∈ V)
367anim1i 340 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝐹 Fn 𝐴𝑥𝐴))
3736adantlr 477 . . . . . . . . . . 11 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐹 Fn 𝐴𝑥𝐴))
38 funfvex 5546 . . . . . . . . . . . 12 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
3938funfni 5330 . . . . . . . . . . 11 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ V)
4037, 39syl 14 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ V)
41 brcog 4808 . . . . . . . . . 10 (((𝐺𝑥) ∈ V ∧ (𝐹𝑥) ∈ V) → ((𝐺𝑥)(𝐹𝐺)(𝐹𝑥) ↔ ∃𝑦((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥))))
4235, 40, 41syl2anc 411 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ((𝐺𝑥)(𝐹𝐺)(𝐹𝑥) ↔ ∃𝑦((𝐺𝑥)𝐺𝑦𝑦𝐹(𝐹𝑥))))
4334, 42mpbird 167 . . . . . . . 8 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐺𝑥)(𝐹𝐺)(𝐹𝑥))
4443adantlr 477 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐺𝑥)(𝐹𝐺)(𝐹𝑥))
45 breq 4019 . . . . . . . 8 ((𝐹𝐺) = ( I ↾ 𝐵) → ((𝐺𝑥)(𝐹𝐺)(𝐹𝑥) ↔ (𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥)))
4645ad2antlr 489 . . . . . . 7 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → ((𝐺𝑥)(𝐹𝐺)(𝐹𝑥) ↔ (𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥)))
4744, 46mpbid 147 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥))
48 fof 5452 . . . . . . . . . 10 (𝐺:𝐴onto𝐵𝐺:𝐴𝐵)
4948adantl 277 . . . . . . . . 9 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐺:𝐴𝐵)
5049ffvelcdmda 5666 . . . . . . . 8 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐺𝑥) ∈ 𝐵)
51 fof 5452 . . . . . . . . . 10 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
5251adantr 276 . . . . . . . . 9 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → 𝐹:𝐴𝐵)
5352ffvelcdmda 5666 . . . . . . . 8 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
54 resieq 4931 . . . . . . . 8 (((𝐺𝑥) ∈ 𝐵 ∧ (𝐹𝑥) ∈ 𝐵) → ((𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥) ↔ (𝐺𝑥) = (𝐹𝑥)))
5550, 53, 54syl2anc 411 . . . . . . 7 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ 𝑥𝐴) → ((𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥) ↔ (𝐺𝑥) = (𝐹𝑥)))
5655adantlr 477 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → ((𝐺𝑥)( I ↾ 𝐵)(𝐹𝑥) ↔ (𝐺𝑥) = (𝐹𝑥)))
5747, 56mpbid 147 . . . . 5 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐺𝑥) = (𝐹𝑥))
5857eqcomd 2194 . . . 4 ((((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
598, 10, 58eqfnfvd 5631 . . 3 (((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) ∧ (𝐹𝐺) = ( I ↾ 𝐵)) → 𝐹 = 𝐺)
6059ex 115 . 2 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → ((𝐹𝐺) = ( I ↾ 𝐵) → 𝐹 = 𝐺))
616, 60impbid 129 1 ((𝐹:𝐴onto𝐵𝐺:𝐴onto𝐵) → (𝐹 = 𝐺 ↔ (𝐹𝐺) = ( I ↾ 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1363  wex 1502  wcel 2159  Vcvv 2751  cop 3609   class class class wbr 4017   I cid 4302  ccnv 4639  cres 4642  ccom 4644   Fn wfn 5225  wf 5226  ontowfo 5228  cfv 5230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-v 2753  df-sbc 2977  df-csb 3072  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-opab 4079  df-mpt 4080  df-id 4307  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-fo 5236  df-fv 5238
This theorem is referenced by: (None)
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