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| Mirrors > Home > MPE Home > Th. List > abrexex2g | Structured version Visualization version GIF version | ||
| Description: Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) |
| Ref | Expression |
|---|---|
| abrexex2g | ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ 𝑊) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑧∃𝑥 ∈ 𝐴 𝜑 | |
| 2 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
| 3 | nfs1v 2156 | . . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑 | |
| 4 | 2, 3 | nfrexw 3313 | . . . 4 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑 |
| 5 | sbequ12 2251 | . . . . 5 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑)) | |
| 6 | 5 | rexbidv 3179 | . . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑)) |
| 7 | 1, 4, 6 | cbvabw 2813 | . . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑} |
| 8 | df-clab 2715 | . . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) | |
| 9 | 8 | rexbii 3094 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑) |
| 10 | 9 | abbii 2809 | . . 3 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑}} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑} |
| 11 | 7, 10 | eqtr4i 2768 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑}} |
| 12 | df-iun 4993 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑}} | |
| 13 | iunexg 7988 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V) | |
| 14 | 12, 13 | eqeltrrid 2846 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ 𝑊) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑}} ∈ V) |
| 15 | 11, 14 | eqeltrid 2845 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ 𝑊) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 [wsb 2064 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ∪ ciun 4991 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-v 3482 df-ss 3968 df-uni 4908 df-iun 4993 |
| This theorem is referenced by: abrexex2 7994 ptrescn 23647 satfvsuclem1 35364 satf0suclem 35380 fmlasuc0 35389 sdclem2 37749 sdclem1 37750 sprval 47466 prprval 47501 |
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