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Theorem eqfnfv3 5635
Description: Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
eqfnfv3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥)))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝑥,𝐵

Proof of Theorem eqfnfv3
StepHypRef Expression
1 eqfnfv2 5634 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
2 eqss 3185 . . . . 5 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
3 ancom 266 . . . . 5 ((𝐴𝐵𝐵𝐴) ↔ (𝐵𝐴𝐴𝐵))
42, 3bitri 184 . . . 4 (𝐴 = 𝐵 ↔ (𝐵𝐴𝐴𝐵))
54anbi1i 458 . . 3 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ ((𝐵𝐴𝐴𝐵) ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
6 anass 401 . . . 4 (((𝐵𝐴𝐴𝐵) ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ (𝐵𝐴 ∧ (𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
7 dfss3 3160 . . . . . . 7 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
87anbi1i 458 . . . . . 6 ((𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ (∀𝑥𝐴 𝑥𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
9 r19.26 2616 . . . . . 6 (∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥)) ↔ (∀𝑥𝐴 𝑥𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
108, 9bitr4i 187 . . . . 5 ((𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥)))
1110anbi2i 457 . . . 4 ((𝐵𝐴 ∧ (𝐴𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))) ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥))))
126, 11bitri 184 . . 3 (((𝐵𝐴𝐴𝐵) ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥))))
135, 12bitri 184 . 2 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)) ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥))))
141, 13bitrdi 196 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  wral 2468  wss 3144   Fn wfn 5230  cfv 5235
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243
This theorem is referenced by: (None)
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