Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mpoexg | Structured version Visualization version GIF version |
Description: Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.) |
Ref | Expression |
---|---|
mpoexg.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Ref | Expression |
---|---|
mpoexg | ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 𝐹 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3511 | . . 3 ⊢ (𝐵 ∈ 𝑆 → 𝐵 ∈ V) | |
2 | elex 3511 | . . . 4 ⊢ (𝐵 ∈ V → 𝐵 ∈ V) | |
3 | 2 | ralrimivw 3182 | . . 3 ⊢ (𝐵 ∈ V → ∀𝑥 ∈ 𝐴 𝐵 ∈ V) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝐵 ∈ 𝑆 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V) |
5 | mpoexg.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
6 | 5 | mpoexxg 7770 | . 2 ⊢ ((𝐴 ∈ 𝑅 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ V) → 𝐹 ∈ V) |
7 | 4, 6 | sylan2 594 | 1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3137 Vcvv 3493 ∈ cmpo 7155 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4836 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-id 5457 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-oprab 7157 df-mpo 7158 df-1st 7686 df-2nd 7687 |
This theorem is referenced by: mpoexga 7772 rmodislmod 19698 eulerpartgbij 31654 hspval 42965 digfval 44731 |
Copyright terms: Public domain | W3C validator |