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Theorem f1veqaeq 5329
Description: If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
Assertion
Ref Expression
f1veqaeq ((𝐹:A1-1B (𝐶 A 𝐷 A)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))

Proof of Theorem f1veqaeq
Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dff13 5328 . . 3 (𝐹:A1-1B ↔ (𝐹:AB 𝑐 A 𝑑 A ((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑)))
2 fveq2 5099 . . . . . . . 8 (𝑐 = 𝐶 → (𝐹𝑐) = (𝐹𝐶))
32eqeq1d 2026 . . . . . . 7 (𝑐 = 𝐶 → ((𝐹𝑐) = (𝐹𝑑) ↔ (𝐹𝐶) = (𝐹𝑑)))
4 eqeq1 2024 . . . . . . 7 (𝑐 = 𝐶 → (𝑐 = 𝑑𝐶 = 𝑑))
53, 4imbi12d 223 . . . . . 6 (𝑐 = 𝐶 → (((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑) ↔ ((𝐹𝐶) = (𝐹𝑑) → 𝐶 = 𝑑)))
6 fveq2 5099 . . . . . . . 8 (𝑑 = 𝐷 → (𝐹𝑑) = (𝐹𝐷))
76eqeq2d 2029 . . . . . . 7 (𝑑 = 𝐷 → ((𝐹𝐶) = (𝐹𝑑) ↔ (𝐹𝐶) = (𝐹𝐷)))
8 eqeq2 2027 . . . . . . 7 (𝑑 = 𝐷 → (𝐶 = 𝑑𝐶 = 𝐷))
97, 8imbi12d 223 . . . . . 6 (𝑑 = 𝐷 → (((𝐹𝐶) = (𝐹𝑑) → 𝐶 = 𝑑) ↔ ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
105, 9rspc2v 2635 . . . . 5 ((𝐶 A 𝐷 A) → (𝑐 A 𝑑 A ((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
1110com12 27 . . . 4 (𝑐 A 𝑑 A ((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑) → ((𝐶 A 𝐷 A) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
1211adantl 262 . . 3 ((𝐹:AB 𝑐 A 𝑑 A ((𝐹𝑐) = (𝐹𝑑) → 𝑐 = 𝑑)) → ((𝐶 A 𝐷 A) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
131, 12sylbi 114 . 2 (𝐹:A1-1B → ((𝐶 A 𝐷 A) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
1413imp 115 1 ((𝐹:A1-1B (𝐶 A 𝐷 A)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1226   wcel 1370  wral 2280  wf 4821  1-1wf1 4822  cfv 4825
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fv 4833
This theorem is referenced by:  f1fveq  5332  f1ocnvfvrneq  5343  f1o2ndf1  5768
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