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Mirrors > Home > MPE Home > Th. List > elmptrab2OLD | Structured version Visualization version GIF version |
Description: Obsolete version of elmptrab2 21442 as of 26-Mar-2021. (Contributed by Stefan O'Rear, 28-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elmptrab2.f | ⊢ 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) |
elmptrab2.s1 | ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓)) |
elmptrab2.s2 | ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶) |
elmptrab2OLD.ex | ⊢ 𝐵 ∈ 𝑉 |
elmptrab2OLD.rc | ⊢ (𝑌 ∈ 𝐶 → 𝑋 ∈ 𝑊) |
Ref | Expression |
---|---|
elmptrab2OLD | ⊢ (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 | elmptrab2OLD.ex | . . . 4 ⊢ 𝐵 ∈ 𝑉 | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑥 ∈ V → 𝐵 ∈ 𝑉) |
6 | 1, 2, 3, 5 | elmptrab 21440 | . 2 ⊢ (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋 ∈ V ∧ 𝑌 ∈ 𝐶 ∧ 𝜓)) |
7 | 3simpc 1053 | . . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ 𝐶 ∧ 𝜓) → (𝑌 ∈ 𝐶 ∧ 𝜓)) | |
8 | elmptrab2OLD.rc | . . . . . 6 ⊢ (𝑌 ∈ 𝐶 → 𝑋 ∈ 𝑊) | |
9 | elex 3185 | . . . . . 6 ⊢ (𝑋 ∈ 𝑊 → 𝑋 ∈ V) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝑌 ∈ 𝐶 → 𝑋 ∈ V) |
11 | 10 | adantr 480 | . . . 4 ⊢ ((𝑌 ∈ 𝐶 ∧ 𝜓) → 𝑋 ∈ V) |
12 | simpl 472 | . . . 4 ⊢ ((𝑌 ∈ 𝐶 ∧ 𝜓) → 𝑌 ∈ 𝐶) | |
13 | simpr 476 | . . . 4 ⊢ ((𝑌 ∈ 𝐶 ∧ 𝜓) → 𝜓) | |
14 | 11, 12, 13 | 3jca 1235 | . . 3 ⊢ ((𝑌 ∈ 𝐶 ∧ 𝜓) → (𝑋 ∈ V ∧ 𝑌 ∈ 𝐶 ∧ 𝜓)) |
15 | 7, 14 | impbii 198 | . 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ 𝐶 ∧ 𝜓) ↔ (𝑌 ∈ 𝐶 ∧ 𝜓)) |
16 | 6, 15 | bitri 263 | 1 ⊢ (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑌 ∈ 𝐶 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ↦ cmpt 4643 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 |
This theorem is referenced by: (None) |
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