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Mirrors > Home > ILE Home > Th. List > mpoxopovel | GIF version |
Description: Element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.) |
Ref | Expression |
---|---|
mpoxopoveq.f | ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑}) |
Ref | Expression |
---|---|
mpoxopovel | ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpoxopoveq.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑}) | |
2 | 1 | mpoxopn0yelv 6243 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 𝐾 ∈ 𝑉)) |
3 | 2 | pm4.71rd 394 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾)))) |
4 | 1 | mpoxopoveq 6244 | . . . . . 6 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (〈𝑉, 𝑊〉𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑}) |
5 | 4 | eleq2d 2247 | . . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) ↔ 𝑁 ∈ {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑})) |
6 | nfcv 2319 | . . . . . . 7 ⊢ Ⅎ𝑛𝑉 | |
7 | 6 | elrabsf 3003 | . . . . . 6 ⊢ (𝑁 ∈ {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑} ↔ (𝑁 ∈ 𝑉 ∧ [𝑁 / 𝑛][〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑)) |
8 | sbccom 3040 | . . . . . . . 8 ⊢ ([𝑁 / 𝑛][〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑 ↔ [〈𝑉, 𝑊〉 / 𝑥][𝑁 / 𝑛][𝐾 / 𝑦]𝜑) | |
9 | sbccom 3040 | . . . . . . . . 9 ⊢ ([𝑁 / 𝑛][𝐾 / 𝑦]𝜑 ↔ [𝐾 / 𝑦][𝑁 / 𝑛]𝜑) | |
10 | 9 | sbcbii 3024 | . . . . . . . 8 ⊢ ([〈𝑉, 𝑊〉 / 𝑥][𝑁 / 𝑛][𝐾 / 𝑦]𝜑 ↔ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑) |
11 | 8, 10 | bitri 184 | . . . . . . 7 ⊢ ([𝑁 / 𝑛][〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑 ↔ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑) |
12 | 11 | anbi2i 457 | . . . . . 6 ⊢ ((𝑁 ∈ 𝑉 ∧ [𝑁 / 𝑛][〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑) ↔ (𝑁 ∈ 𝑉 ∧ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)) |
13 | 7, 12 | bitri 184 | . . . . 5 ⊢ (𝑁 ∈ {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑} ↔ (𝑁 ∈ 𝑉 ∧ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)) |
14 | 5, 13 | bitrdi 196 | . . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) ↔ (𝑁 ∈ 𝑉 ∧ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))) |
15 | 14 | pm5.32da 452 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → ((𝐾 ∈ 𝑉 ∧ 𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾)) ↔ (𝐾 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))) |
16 | 3anass 982 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑) ↔ (𝐾 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))) | |
17 | 15, 16 | bitr4di 198 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → ((𝐾 ∈ 𝑉 ∧ 𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾)) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))) |
18 | 3, 17 | bitrd 188 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 {crab 2459 Vcvv 2739 [wsbc 2964 〈cop 3597 ‘cfv 5218 (class class class)co 5878 ∈ cmpo 5880 1st c1st 6142 |
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 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 |
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 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fv 5226 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 |
This theorem is referenced by: (None) |
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