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Mirrors > Home > MPE Home > Th. List > elmptrab2 | Structured version Visualization version GIF version |
Description: Membership in a one-parameter class of sets, indexed by arbitrary base sets. (Contributed by Stefan O'Rear, 28-Jul-2015.) (Revised by AV, 26-Mar-2021.) |
Ref | Expression |
---|---|
elmptrab2.f | ⊢ 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) |
elmptrab2.s1 | ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓)) |
elmptrab2.s2 | ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶) |
elmptrab2.ex | ⊢ 𝐵 ∈ V |
elmptrab2.rc | ⊢ (𝑌 ∈ 𝐶 → 𝑋 ∈ 𝑊) |
Ref | Expression |
---|---|
elmptrab2 | ⊢ (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑌 ∈ 𝐶 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmptrab2.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) | |
2 | elmptrab2.s1 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓)) | |
3 | elmptrab2.s2 | . . 3 ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶) | |
4 | elmptrab2.ex | . . . 4 ⊢ 𝐵 ∈ V | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑥 ∈ V → 𝐵 ∈ V) |
6 | 1, 2, 3, 5 | elmptrab 22432 | . 2 ⊢ (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋 ∈ V ∧ 𝑌 ∈ 𝐶 ∧ 𝜓)) |
7 | 3simpc 1147 | . . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ 𝐶 ∧ 𝜓) → (𝑌 ∈ 𝐶 ∧ 𝜓)) | |
8 | elmptrab2.rc | . . . . . 6 ⊢ (𝑌 ∈ 𝐶 → 𝑋 ∈ 𝑊) | |
9 | 8 | elexd 3461 | . . . . 5 ⊢ (𝑌 ∈ 𝐶 → 𝑋 ∈ V) |
10 | 9 | adantr 484 | . . . 4 ⊢ ((𝑌 ∈ 𝐶 ∧ 𝜓) → 𝑋 ∈ V) |
11 | simpl 486 | . . . 4 ⊢ ((𝑌 ∈ 𝐶 ∧ 𝜓) → 𝑌 ∈ 𝐶) | |
12 | simpr 488 | . . . 4 ⊢ ((𝑌 ∈ 𝐶 ∧ 𝜓) → 𝜓) | |
13 | 10, 11, 12 | 3jca 1125 | . . 3 ⊢ ((𝑌 ∈ 𝐶 ∧ 𝜓) → (𝑋 ∈ V ∧ 𝑌 ∈ 𝐶 ∧ 𝜓)) |
14 | 7, 13 | impbii 212 | . 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ 𝐶 ∧ 𝜓) ↔ (𝑌 ∈ 𝐶 ∧ 𝜓)) |
15 | 6, 14 | bitri 278 | 1 ⊢ (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑌 ∈ 𝐶 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ↦ cmpt 5110 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fv 6332 |
This theorem is referenced by: isfil 22452 isufil 22508 |
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