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| Mirrors > Home > MPE Home > Th. List > eqfunfv | Structured version Visualization version GIF version | ||
| Description: Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.) |
| Ref | Expression |
|---|---|
| eqfunfv | ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 = 𝐺 ↔ (dom 𝐹 = dom 𝐺 ∧ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) = (𝐺‘𝑥)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | funfn 6596 | . 2 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
| 2 | funfn 6596 | . 2 ⊢ (Fun 𝐺 ↔ 𝐺 Fn dom 𝐺) | |
| 3 | eqfnfv2 7052 | . 2 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐺 Fn dom 𝐺) → (𝐹 = 𝐺 ↔ (dom 𝐹 = dom 𝐺 ∧ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) = (𝐺‘𝑥)))) | |
| 4 | 1, 2, 3 | syl2anb 598 | 1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 = 𝐺 ↔ (dom 𝐹 = dom 𝐺 ∧ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) = (𝐺‘𝑥)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∀wral 3061 dom cdm 5685 Fun wfun 6555 Fn wfn 6556 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: fveqressseq 7099 fnprb 7228 fntpb 7229 symgfixf1 19455 nosepon 27710 nolesgn2ores 27717 nogesgn1ores 27719 nosupres 27752 nosupbnd2lem1 27760 noinfres 27767 noinfbnd2lem1 27775 noetasuplem4 27781 noetainflem4 27785 comptiunov2i 43719 |
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