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Mirrors > Home > MPE Home > Th. List > rexrnmpt2 | Structured version Visualization version GIF version |
Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.) |
Ref | Expression |
---|---|
rngop.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
ralrnmpt2.2 | ⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rexrnmpt2 | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngop.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
2 | ralrnmpt2.2 | . . . . 5 ⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓)) | |
3 | 2 | notbid 307 | . . . 4 ⊢ (𝑧 = 𝐶 → (¬ 𝜑 ↔ ¬ 𝜓)) |
4 | 1, 3 | ralrnmpt2 6673 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑧 ∈ ran 𝐹 ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜓)) |
5 | 4 | notbid 307 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (¬ ∀𝑧 ∈ ran 𝐹 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜓)) |
6 | dfrex2 2979 | . 2 ⊢ (∃𝑧 ∈ ran 𝐹𝜑 ↔ ¬ ∀𝑧 ∈ ran 𝐹 ¬ 𝜑) | |
7 | dfrex2 2979 | . . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜓) | |
8 | 7 | rexbii 3023 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜓) |
9 | rexnal 2978 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜓) | |
10 | 8, 9 | bitri 263 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜓) |
11 | 5, 6, 10 | 3bitr4g 302 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ran crn 5039 ↦ cmpt2 6551 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 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-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-br 4584 df-opab 4644 df-cnv 5046 df-dm 5048 df-rn 5049 df-oprab 6553 df-mpt2 6554 |
This theorem is referenced by: lsmass 17906 eltx 21181 txrest 21244 txlm 21261 ptrest 32578 |
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