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Mirrors > Home > ILE Home > Th. List > elpmg | GIF version |
Description: The predicate "is a partial function". (Contributed by Mario Carneiro, 14-Nov-2013.) |
Ref | Expression |
---|---|
elpmg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝐶 ∧ 𝐶 ⊆ (𝐵 × 𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmvalg 6659 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ↑pm 𝐵) = {𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑔}) | |
2 | 1 | eleq2d 2247 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑pm 𝐵) ↔ 𝐶 ∈ {𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑔})) |
3 | funeq 5237 | . . . . 5 ⊢ (𝑔 = 𝐶 → (Fun 𝑔 ↔ Fun 𝐶)) | |
4 | 3 | elrab 2894 | . . . 4 ⊢ (𝐶 ∈ {𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑔} ↔ (𝐶 ∈ 𝒫 (𝐵 × 𝐴) ∧ Fun 𝐶)) |
5 | 2, 4 | bitrdi 196 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑pm 𝐵) ↔ (𝐶 ∈ 𝒫 (𝐵 × 𝐴) ∧ Fun 𝐶))) |
6 | ancom 266 | . . 3 ⊢ ((𝐶 ∈ 𝒫 (𝐵 × 𝐴) ∧ Fun 𝐶) ↔ (Fun 𝐶 ∧ 𝐶 ∈ 𝒫 (𝐵 × 𝐴))) | |
7 | 5, 6 | bitrdi 196 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝐶 ∧ 𝐶 ∈ 𝒫 (𝐵 × 𝐴)))) |
8 | elex 2749 | . . . . 5 ⊢ (𝐶 ∈ 𝒫 (𝐵 × 𝐴) → 𝐶 ∈ V) | |
9 | 8 | a1i 9 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ 𝒫 (𝐵 × 𝐴) → 𝐶 ∈ V)) |
10 | xpexg 4741 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 × 𝐴) ∈ V) | |
11 | 10 | ancoms 268 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × 𝐴) ∈ V) |
12 | ssexg 4143 | . . . . . 6 ⊢ ((𝐶 ⊆ (𝐵 × 𝐴) ∧ (𝐵 × 𝐴) ∈ V) → 𝐶 ∈ V) | |
13 | 12 | expcom 116 | . . . . 5 ⊢ ((𝐵 × 𝐴) ∈ V → (𝐶 ⊆ (𝐵 × 𝐴) → 𝐶 ∈ V)) |
14 | 11, 13 | syl 14 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ⊆ (𝐵 × 𝐴) → 𝐶 ∈ V)) |
15 | elpwg 3584 | . . . . 5 ⊢ (𝐶 ∈ V → (𝐶 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝐶 ⊆ (𝐵 × 𝐴))) | |
16 | 15 | a1i 9 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ V → (𝐶 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝐶 ⊆ (𝐵 × 𝐴)))) |
17 | 9, 14, 16 | pm5.21ndd 705 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝐶 ⊆ (𝐵 × 𝐴))) |
18 | 17 | anbi2d 464 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((Fun 𝐶 ∧ 𝐶 ∈ 𝒫 (𝐵 × 𝐴)) ↔ (Fun 𝐶 ∧ 𝐶 ⊆ (𝐵 × 𝐴)))) |
19 | 7, 18 | bitrd 188 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝐶 ∧ 𝐶 ⊆ (𝐵 × 𝐴)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2148 {crab 2459 Vcvv 2738 ⊆ wss 3130 𝒫 cpw 3576 × cxp 4625 Fun wfun 5211 (class class class)co 5875 ↑pm cpm 6649 |
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-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2740 df-sbc 2964 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-br 4005 df-opab 4066 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-iota 5179 df-fun 5219 df-fv 5225 df-ov 5878 df-oprab 5879 df-mpo 5880 df-pm 6651 |
This theorem is referenced by: elpm2g 6665 pmss12g 6675 elpm 6679 pmsspw 6683 ennnfonelemj0 12402 lmfss 13747 |
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