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Mirrors > Home > MPE Home > Th. List > ovmpt2ga | Structured version Visualization version GIF version |
Description: Value of an operation given by a maps-to rule. (Contributed by Mario Carneiro, 19-Dec-2013.) |
Ref | Expression |
---|---|
ovmpt2ga.1 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) |
ovmpt2ga.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) |
Ref | Expression |
---|---|
ovmpt2ga | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3185 | . 2 ⊢ (𝑆 ∈ 𝐻 → 𝑆 ∈ V) | |
2 | ovmpt2ga.2 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V) → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)) |
4 | ovmpt2ga.1 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) | |
5 | 4 | adantl 481 | . . 3 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) |
6 | simp1 1054 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V) → 𝐴 ∈ 𝐶) | |
7 | simp2 1055 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V) → 𝐵 ∈ 𝐷) | |
8 | simp3 1056 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V) → 𝑆 ∈ V) | |
9 | 3, 5, 6, 7, 8 | ovmpt2d 6686 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆) |
10 | 1, 9 | syl3an3 1353 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 (class class class)co 6549 ↦ cmpt2 6551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 |
This theorem is referenced by: ovmpt2a 6689 ovmpt2g 6693 elovmpt2 6777 offval 6802 offval3 7053 bropopvvv 7142 reps 13368 hashbcval 15544 setsvalg 15719 ressval 15754 restval 15910 sylow1lem4 17839 sylow3lem2 17866 sylow3lem3 17867 lsmvalx 17877 mvrfval 19241 opsrval 19295 marrepfval 20185 marrepval0 20186 marepvfval 20190 marepvval0 20191 cnmpt12 21280 cnmpt22 21287 qtopval 21308 flimval 21577 fclsval 21622 ucnval 21891 stdbdmetval 22129 wlkon 26061 trlon 26070 pthon 26105 spthon 26112 is2wlkonot 26390 is2spthonot 26391 2wlkonot 26392 2spthonot 26393 2wlksot 26394 2spthsot 26395 2wlkonot3v 26402 2spthonot3v 26403 resvval 29158 ofcfval3 29491 fmulcl 38648 mptmpt2opabbrd 40335 |
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