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Mirrors > Home > ILE Home > Th. List > mpoexg | 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 2762 | . . 3 ⊢ (𝐵 ∈ 𝑆 → 𝐵 ∈ V) | |
2 | elex 2762 | . . . 4 ⊢ (𝐵 ∈ V → 𝐵 ∈ V) | |
3 | 2 | ralrimivw 2563 | . . 3 ⊢ (𝐵 ∈ V → ∀𝑥 ∈ 𝐴 𝐵 ∈ V) |
4 | 1, 3 | syl 14 | . 2 ⊢ (𝐵 ∈ 𝑆 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V) |
5 | mpoexg.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
6 | 5 | mpoexxg 6228 | . 2 ⊢ ((𝐴 ∈ 𝑅 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ V) → 𝐹 ∈ V) |
7 | 4, 6 | sylan2 286 | 1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 𝐹 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1363 ∈ wcel 2159 ∀wral 2467 Vcvv 2751 ∈ cmpo 5892 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-coll 4132 ax-sep 4135 ax-pow 4188 ax-pr 4223 ax-un 4447 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ral 2472 df-rex 2473 df-reu 2474 df-rab 2476 df-v 2753 df-sbc 2977 df-csb 3072 df-un 3147 df-in 3149 df-ss 3156 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-iun 3902 df-br 4018 df-opab 4079 df-mpt 4080 df-id 4307 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-rn 4651 df-res 4652 df-ima 4653 df-iota 5192 df-fun 5232 df-fn 5233 df-f 5234 df-f1 5235 df-fo 5236 df-f1o 5237 df-fv 5238 df-oprab 5894 df-mpo 5895 df-1st 6158 df-2nd 6159 |
This theorem is referenced by: mpoexga 6230 xpsval 12793 rmodislmod 13627 |
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