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Mirrors > Home > MPE Home > Th. List > mpt2xopoveqd | Structured version Visualization version GIF version |
Description: 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, deduction version. (Contributed by Alexander van der Vekens, 11-Oct-2017.) |
Ref | Expression |
---|---|
mpt2xopoveq.f | ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑}) |
mpt2xopoveqd.1 | ⊢ (𝜓 → (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) |
mpt2xopoveqd.2 | ⊢ ((𝜓 ∧ ¬ 𝐾 ∈ 𝑉) → {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑} = ∅) |
Ref | Expression |
---|---|
mpt2xopoveqd | ⊢ (𝜓 → (〈𝑉, 𝑊〉𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpt2xopoveq.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑}) | |
2 | 1 | mpt2xopoveq 7345 | . . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (〈𝑉, 𝑊〉𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑}) |
3 | 2 | ex 450 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐾 ∈ 𝑉 → (〈𝑉, 𝑊〉𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑})) |
4 | mpt2xopoveqd.1 | . . 3 ⊢ (𝜓 → (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) | |
5 | 3, 4 | syl11 33 | . 2 ⊢ (𝐾 ∈ 𝑉 → (𝜓 → (〈𝑉, 𝑊〉𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑})) |
6 | df-nel 2898 | . . . . . 6 ⊢ (𝐾 ∉ 𝑉 ↔ ¬ 𝐾 ∈ 𝑉) | |
7 | 1 | mpt2xopynvov0 7344 | . . . . . 6 ⊢ (𝐾 ∉ 𝑉 → (〈𝑉, 𝑊〉𝐹𝐾) = ∅) |
8 | 6, 7 | sylbir 225 | . . . . 5 ⊢ (¬ 𝐾 ∈ 𝑉 → (〈𝑉, 𝑊〉𝐹𝐾) = ∅) |
9 | 8 | adantr 481 | . . . 4 ⊢ ((¬ 𝐾 ∈ 𝑉 ∧ 𝜓) → (〈𝑉, 𝑊〉𝐹𝐾) = ∅) |
10 | mpt2xopoveqd.2 | . . . . . 6 ⊢ ((𝜓 ∧ ¬ 𝐾 ∈ 𝑉) → {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑} = ∅) | |
11 | 10 | eqcomd 2628 | . . . . 5 ⊢ ((𝜓 ∧ ¬ 𝐾 ∈ 𝑉) → ∅ = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑}) |
12 | 11 | ancoms 469 | . . . 4 ⊢ ((¬ 𝐾 ∈ 𝑉 ∧ 𝜓) → ∅ = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑}) |
13 | 9, 12 | eqtrd 2656 | . . 3 ⊢ ((¬ 𝐾 ∈ 𝑉 ∧ 𝜓) → (〈𝑉, 𝑊〉𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑}) |
14 | 13 | ex 450 | . 2 ⊢ (¬ 𝐾 ∈ 𝑉 → (𝜓 → (〈𝑉, 𝑊〉𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑})) |
15 | 5, 14 | pm2.61i 176 | 1 ⊢ (𝜓 → (〈𝑉, 𝑊〉𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∉ wnel 2897 {crab 2916 Vcvv 3200 [wsbc 3435 ∅c0 3915 〈cop 4183 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1st c1st 7166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 |
This theorem is referenced by: (None) |
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