Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > abfmpunirn | Structured version Visualization version GIF version |
Description: Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 28-Sep-2016.) |
Ref | Expression |
---|---|
abfmpunirn.1 | ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜑}) |
abfmpunirn.2 | ⊢ {𝑦 ∣ 𝜑} ∈ V |
abfmpunirn.3 | ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
abfmpunirn | ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥 ∈ 𝑉 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3512 | . 2 ⊢ (𝐵 ∈ ∪ ran 𝐹 → 𝐵 ∈ V) | |
2 | abfmpunirn.2 | . . . . . 6 ⊢ {𝑦 ∣ 𝜑} ∈ V | |
3 | abfmpunirn.1 | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜑}) | |
4 | 2, 3 | fnmpti 6491 | . . . . 5 ⊢ 𝐹 Fn 𝑉 |
5 | fnunirn 7012 | . . . . 5 ⊢ (𝐹 Fn 𝑉 → (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 𝐵 ∈ (𝐹‘𝑥))) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 𝐵 ∈ (𝐹‘𝑥)) |
7 | 3 | fvmpt2 6779 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝑉 ∧ {𝑦 ∣ 𝜑} ∈ V) → (𝐹‘𝑥) = {𝑦 ∣ 𝜑}) |
8 | 2, 7 | mpan2 689 | . . . . . 6 ⊢ (𝑥 ∈ 𝑉 → (𝐹‘𝑥) = {𝑦 ∣ 𝜑}) |
9 | 8 | eleq2d 2898 | . . . . 5 ⊢ (𝑥 ∈ 𝑉 → (𝐵 ∈ (𝐹‘𝑥) ↔ 𝐵 ∈ {𝑦 ∣ 𝜑})) |
10 | 9 | rexbiia 3246 | . . . 4 ⊢ (∃𝑥 ∈ 𝑉 𝐵 ∈ (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝑉 𝐵 ∈ {𝑦 ∣ 𝜑}) |
11 | 6, 10 | bitri 277 | . . 3 ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 𝐵 ∈ {𝑦 ∣ 𝜑}) |
12 | abfmpunirn.3 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓)) | |
13 | 12 | elabg 3666 | . . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∈ {𝑦 ∣ 𝜑} ↔ 𝜓)) |
14 | 13 | rexbidv 3297 | . . 3 ⊢ (𝐵 ∈ V → (∃𝑥 ∈ 𝑉 𝐵 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝑉 𝜓)) |
15 | 11, 14 | syl5bb 285 | . 2 ⊢ (𝐵 ∈ V → (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 𝜓)) |
16 | 1, 15 | biadanii 820 | 1 ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥 ∈ 𝑉 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2799 ∃wrex 3139 Vcvv 3494 ∪ cuni 4838 ↦ cmpt 5146 ran crn 5556 Fn wfn 6350 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-fv 6363 |
This theorem is referenced by: rabfmpunirn 30398 isrnsiga 31372 isrnmeas 31459 |
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