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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 3500 | . 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 6710 | . . . . 5 ⊢ 𝐹 Fn 𝑉 | 
| 5 | fnunirn 7275 | . . . . 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 7026 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝑉 ∧ {𝑦 ∣ 𝜑} ∈ V) → (𝐹‘𝑥) = {𝑦 ∣ 𝜑}) | 
| 8 | 2, 7 | mpan2 691 | . . . . . 6 ⊢ (𝑥 ∈ 𝑉 → (𝐹‘𝑥) = {𝑦 ∣ 𝜑}) | 
| 9 | 8 | eleq2d 2826 | . . . . 5 ⊢ (𝑥 ∈ 𝑉 → (𝐵 ∈ (𝐹‘𝑥) ↔ 𝐵 ∈ {𝑦 ∣ 𝜑})) | 
| 10 | 9 | rexbiia 3091 | . . . 4 ⊢ (∃𝑥 ∈ 𝑉 𝐵 ∈ (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝑉 𝐵 ∈ {𝑦 ∣ 𝜑}) | 
| 11 | 6, 10 | bitri 275 | . . 3 ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 𝐵 ∈ {𝑦 ∣ 𝜑}) | 
| 12 | abfmpunirn.3 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓)) | |
| 13 | 12 | elabg 3675 | . . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∈ {𝑦 ∣ 𝜑} ↔ 𝜓)) | 
| 14 | 13 | rexbidv 3178 | . . 3 ⊢ (𝐵 ∈ V → (∃𝑥 ∈ 𝑉 𝐵 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝑉 𝜓)) | 
| 15 | 11, 14 | bitrid 283 | . 2 ⊢ (𝐵 ∈ V → (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 𝜓)) | 
| 16 | 1, 15 | biadanii 821 | 1 ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥 ∈ 𝑉 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {cab 2713 ∃wrex 3069 Vcvv 3479 ∪ cuni 4906 ↦ cmpt 5224 ran crn 5685 Fn wfn 6555 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-fv 6568 | 
| This theorem is referenced by: rabfmpunirn 32664 isrnsiga 34115 isrnmeas 34202 | 
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