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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 23816 | . 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 3485 | . . . . 5 ⊢ (𝑌 ∈ 𝐶 → 𝑋 ∈ V) |
10 | 9 | adantr 479 | . . . 4 ⊢ ((𝑌 ∈ 𝐶 ∧ 𝜓) → 𝑋 ∈ V) |
11 | simpl 481 | . . . 4 ⊢ ((𝑌 ∈ 𝐶 ∧ 𝜓) → 𝑌 ∈ 𝐶) | |
12 | simpr 483 | . . . 4 ⊢ ((𝑌 ∈ 𝐶 ∧ 𝜓) → 𝜓) | |
13 | 10, 11, 12 | 3jca 1125 | . . 3 ⊢ ((𝑌 ∈ 𝐶 ∧ 𝜓) → (𝑋 ∈ V ∧ 𝑌 ∈ 𝐶 ∧ 𝜓)) |
14 | 7, 13 | impbii 208 | . 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ 𝐶 ∧ 𝜓) ↔ (𝑌 ∈ 𝐶 ∧ 𝜓)) |
15 | 6, 14 | bitri 274 | 1 ⊢ (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑌 ∈ 𝐶 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 {crab 3419 Vcvv 3462 ↦ cmpt 5226 ‘cfv 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fv 6551 |
This theorem is referenced by: isfil 23836 isufil 23892 |
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