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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 22438 | . 2 ⊢ (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋 ∈ V ∧ 𝑌 ∈ 𝐶 ∧ 𝜓)) |
7 | 3simpc 1146 | . . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ 𝐶 ∧ 𝜓) → (𝑌 ∈ 𝐶 ∧ 𝜓)) | |
8 | elmptrab2.rc | . . . . . 6 ⊢ (𝑌 ∈ 𝐶 → 𝑋 ∈ 𝑊) | |
9 | 8 | elexd 3517 | . . . . 5 ⊢ (𝑌 ∈ 𝐶 → 𝑋 ∈ V) |
10 | 9 | adantr 483 | . . . 4 ⊢ ((𝑌 ∈ 𝐶 ∧ 𝜓) → 𝑋 ∈ V) |
11 | simpl 485 | . . . 4 ⊢ ((𝑌 ∈ 𝐶 ∧ 𝜓) → 𝑌 ∈ 𝐶) | |
12 | simpr 487 | . . . 4 ⊢ ((𝑌 ∈ 𝐶 ∧ 𝜓) → 𝜓) | |
13 | 10, 11, 12 | 3jca 1124 | . . 3 ⊢ ((𝑌 ∈ 𝐶 ∧ 𝜓) → (𝑋 ∈ V ∧ 𝑌 ∈ 𝐶 ∧ 𝜓)) |
14 | 7, 13 | impbii 211 | . 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ 𝐶 ∧ 𝜓) ↔ (𝑌 ∈ 𝐶 ∧ 𝜓)) |
15 | 6, 14 | bitri 277 | 1 ⊢ (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑌 ∈ 𝐶 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 {crab 3145 Vcvv 3497 ↦ cmpt 5149 ‘cfv 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fv 6366 |
This theorem is referenced by: isfil 22458 isufil 22514 |
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