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Theorem cocan2 5371
Description: A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
cocan2 ((𝐹:AontoB 𝐻 Fn B 𝐾 Fn B) → ((𝐻𝐹) = (𝐾𝐹) ↔ 𝐻 = 𝐾))

Proof of Theorem cocan2
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fof 5049 . . . . . . 7 (𝐹:AontoB𝐹:AB)
213ad2ant1 924 . . . . . 6 ((𝐹:AontoB 𝐻 Fn B 𝐾 Fn B) → 𝐹:AB)
3 fvco3 5187 . . . . . 6 ((𝐹:AB y A) → ((𝐻𝐹)‘y) = (𝐻‘(𝐹y)))
42, 3sylan 267 . . . . 5 (((𝐹:AontoB 𝐻 Fn B 𝐾 Fn B) y A) → ((𝐻𝐹)‘y) = (𝐻‘(𝐹y)))
5 fvco3 5187 . . . . . 6 ((𝐹:AB y A) → ((𝐾𝐹)‘y) = (𝐾‘(𝐹y)))
62, 5sylan 267 . . . . 5 (((𝐹:AontoB 𝐻 Fn B 𝐾 Fn B) y A) → ((𝐾𝐹)‘y) = (𝐾‘(𝐹y)))
74, 6eqeq12d 2051 . . . 4 (((𝐹:AontoB 𝐻 Fn B 𝐾 Fn B) y A) → (((𝐻𝐹)‘y) = ((𝐾𝐹)‘y) ↔ (𝐻‘(𝐹y)) = (𝐾‘(𝐹y))))
87ralbidva 2316 . . 3 ((𝐹:AontoB 𝐻 Fn B 𝐾 Fn B) → (y A ((𝐻𝐹)‘y) = ((𝐾𝐹)‘y) ↔ y A (𝐻‘(𝐹y)) = (𝐾‘(𝐹y))))
9 fveq2 5121 . . . . . 6 ((𝐹y) = x → (𝐻‘(𝐹y)) = (𝐻x))
10 fveq2 5121 . . . . . 6 ((𝐹y) = x → (𝐾‘(𝐹y)) = (𝐾x))
119, 10eqeq12d 2051 . . . . 5 ((𝐹y) = x → ((𝐻‘(𝐹y)) = (𝐾‘(𝐹y)) ↔ (𝐻x) = (𝐾x)))
1211cbvfo 5368 . . . 4 (𝐹:AontoB → (y A (𝐻‘(𝐹y)) = (𝐾‘(𝐹y)) ↔ x B (𝐻x) = (𝐾x)))
13123ad2ant1 924 . . 3 ((𝐹:AontoB 𝐻 Fn B 𝐾 Fn B) → (y A (𝐻‘(𝐹y)) = (𝐾‘(𝐹y)) ↔ x B (𝐻x) = (𝐾x)))
148, 13bitrd 177 . 2 ((𝐹:AontoB 𝐻 Fn B 𝐾 Fn B) → (y A ((𝐻𝐹)‘y) = ((𝐾𝐹)‘y) ↔ x B (𝐻x) = (𝐾x)))
15 simp2 904 . . . 4 ((𝐹:AontoB 𝐻 Fn B 𝐾 Fn B) → 𝐻 Fn B)
16 fnfco 5008 . . . 4 ((𝐻 Fn B 𝐹:AB) → (𝐻𝐹) Fn A)
1715, 2, 16syl2anc 391 . . 3 ((𝐹:AontoB 𝐻 Fn B 𝐾 Fn B) → (𝐻𝐹) Fn A)
18 simp3 905 . . . 4 ((𝐹:AontoB 𝐻 Fn B 𝐾 Fn B) → 𝐾 Fn B)
19 fnfco 5008 . . . 4 ((𝐾 Fn B 𝐹:AB) → (𝐾𝐹) Fn A)
2018, 2, 19syl2anc 391 . . 3 ((𝐹:AontoB 𝐻 Fn B 𝐾 Fn B) → (𝐾𝐹) Fn A)
21 eqfnfv 5208 . . 3 (((𝐻𝐹) Fn A (𝐾𝐹) Fn A) → ((𝐻𝐹) = (𝐾𝐹) ↔ y A ((𝐻𝐹)‘y) = ((𝐾𝐹)‘y)))
2217, 20, 21syl2anc 391 . 2 ((𝐹:AontoB 𝐻 Fn B 𝐾 Fn B) → ((𝐻𝐹) = (𝐾𝐹) ↔ y A ((𝐻𝐹)‘y) = ((𝐾𝐹)‘y)))
23 eqfnfv 5208 . . 3 ((𝐻 Fn B 𝐾 Fn B) → (𝐻 = 𝐾x B (𝐻x) = (𝐾x)))
2415, 18, 23syl2anc 391 . 2 ((𝐹:AontoB 𝐻 Fn B 𝐾 Fn B) → (𝐻 = 𝐾x B (𝐻x) = (𝐾x)))
2514, 22, 243bitr4d 209 1 ((𝐹:AontoB 𝐻 Fn B 𝐾 Fn B) → ((𝐻𝐹) = (𝐾𝐹) ↔ 𝐻 = 𝐾))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  wral 2300  ccom 4292   Fn wfn 4840  wf 4841  ontowfo 4843  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853
This theorem is referenced by: (None)
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