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Mirrors > Home > MPE Home > Th. List > eqfnov2 | Structured version Visualization version GIF version |
Description: Two operators with the same domain are equal iff their values at each point in the domain are equal. (Contributed by Jeff Madsen, 7-Jun-2010.) |
Ref | Expression |
---|---|
eqfnov2 | ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐴 × 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqfnov 7269 | . 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐴 × 𝐵)) → (𝐹 = 𝐺 ↔ ((𝐴 × 𝐵) = (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))) | |
2 | simpr 485 | . . 3 ⊢ (((𝐴 × 𝐵) = (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) | |
3 | eqidd 2819 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦) → (𝐴 × 𝐵) = (𝐴 × 𝐵)) | |
4 | 3 | ancri 550 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦) → ((𝐴 × 𝐵) = (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))) |
5 | 2, 4 | impbii 210 | . 2 ⊢ (((𝐴 × 𝐵) = (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) |
6 | 1, 5 | syl6bb 288 | 1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐴 × 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∀wral 3135 × cxp 5546 Fn wfn 6343 (class class class)co 7145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 df-ov 7148 |
This theorem is referenced by: fnmpoovd 7771 tpossym 7913 uncfcurf 17477 mamuass 20939 mamudi 20940 mamudir 20941 mamuvs1 20942 mamuvs2 20943 eqmat 20961 mamulid 20978 mamurid 20979 madutpos 21179 ressprdsds 22908 isngp3 23134 xrsdsre 23345 hhip 28881 1smat1 30968 matunitlindflem2 34770 |
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