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Theorem eqfnfv3 5210
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 5209 . 2 ((𝐹 Fn A 𝐺 Fn B) → (𝐹 = 𝐺 ↔ (A = B x A (𝐹x) = (𝐺x))))
2 eqss 2954 . . . . 5 (A = B ↔ (AB BA))
3 ancom 253 . . . . 5 ((AB BA) ↔ (BA AB))
42, 3bitri 173 . . . 4 (A = B ↔ (BA AB))
54anbi1i 431 . . 3 ((A = B x A (𝐹x) = (𝐺x)) ↔ ((BA AB) x A (𝐹x) = (𝐺x)))
6 anass 381 . . . 4 (((BA AB) x A (𝐹x) = (𝐺x)) ↔ (BA (AB x A (𝐹x) = (𝐺x))))
7 dfss3 2929 . . . . . . 7 (ABx A x B)
87anbi1i 431 . . . . . 6 ((AB x A (𝐹x) = (𝐺x)) ↔ (x A x B x A (𝐹x) = (𝐺x)))
9 r19.26 2435 . . . . . 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 430 . . . 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 1242   wcel 1390  wral 2300  wss 2911   Fn wfn 4840  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-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by: (None)
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