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Theorem eqfnfv3 5192
Description: Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
eqfnfv3 ((𝐹 Fn A 𝐺 Fn B) → (𝐹 = 𝐺 ↔ (BA x A (x B (𝐹x) = (𝐺x)))))
Distinct variable groups:   x,A   x,𝐹   x,𝐺   x,B

Proof of Theorem eqfnfv3
StepHypRef Expression
1 eqfnfv2 5191 . 2 ((𝐹 Fn A 𝐺 Fn B) → (𝐹 = 𝐺 ↔ (A = B x A (𝐹x) = (𝐺x))))
2 eqss 2937 . . . . 5 (A = B ↔ (AB BA))
3 ancom 253 . . . . 5 ((AB BA) ↔ (BA AB))
42, 3bitri 173 . . . 4 (A = B ↔ (BA AB))
54anbi1i 434 . . 3 ((A = B x A (𝐹x) = (𝐺x)) ↔ ((BA AB) x A (𝐹x) = (𝐺x)))
6 anass 383 . . . 4 (((BA AB) x A (𝐹x) = (𝐺x)) ↔ (BA (AB x A (𝐹x) = (𝐺x))))
7 dfss3 2912 . . . . . . 7 (ABx A x B)
87anbi1i 434 . . . . . 6 ((AB x A (𝐹x) = (𝐺x)) ↔ (x A x B x A (𝐹x) = (𝐺x)))
9 r19.26 2419 . . . . . 6 (x A (x B (𝐹x) = (𝐺x)) ↔ (x A x B x A (𝐹x) = (𝐺x)))
108, 9bitr4i 176 . . . . 5 ((AB x A (𝐹x) = (𝐺x)) ↔ x A (x B (𝐹x) = (𝐺x)))
1110anbi2i 433 . . . 4 ((BA (AB x A (𝐹x) = (𝐺x))) ↔ (BA x A (x B (𝐹x) = (𝐺x))))
126, 11bitri 173 . . 3 (((BA AB) x A (𝐹x) = (𝐺x)) ↔ (BA x A (x B (𝐹x) = (𝐺x))))
135, 12bitri 173 . 2 ((A = B x A (𝐹x) = (𝐺x)) ↔ (BA x A (x B (𝐹x) = (𝐺x))))
141, 13syl6bb 185 1 ((𝐹 Fn A 𝐺 Fn B) → (𝐹 = 𝐺 ↔ (BA x A (x B (𝐹x) = (𝐺x)))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1374  wral 2284  wss 2894   Fn wfn 4824  cfv 4829
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by: (None)
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