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Mirrors > Home > MPE Home > Th. List > elabrex | Structured version Visualization version GIF version |
Description: Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.) |
Ref | Expression |
---|---|
elabrex.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
elabrex | ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1661 | . . . 4 ⊢ ⊤ | |
2 | csbeq1a 3766 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) | |
3 | 2 | equcoms 2124 | . . . . . 6 ⊢ (𝑧 = 𝑥 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) |
4 | trud 1667 | . . . . . 6 ⊢ (𝑧 = 𝑥 → ⊤) | |
5 | 3, 4 | 2thd 257 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝐵 = ⦋𝑧 / 𝑥⦌𝐵 ↔ ⊤)) |
6 | 5 | rspcev 3526 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ⊤) → ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) |
7 | 1, 6 | mpan2 682 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) |
8 | elabrex.1 | . . . 4 ⊢ 𝐵 ∈ V | |
9 | eqeq1 2829 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 = ⦋𝑧 / 𝑥⦌𝐵 ↔ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)) | |
10 | 9 | rexbidv 3262 | . . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵 ↔ ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)) |
11 | 8, 10 | elab 3571 | . . 3 ⊢ (𝐵 ∈ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵} ↔ ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) |
12 | 7, 11 | sylibr 226 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵}) |
13 | nfv 2013 | . . . 4 ⊢ Ⅎ𝑧 𝑦 = 𝐵 | |
14 | nfcsb1v 3773 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵 | |
15 | 14 | nfeq2 2985 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = ⦋𝑧 / 𝑥⦌𝐵 |
16 | 2 | eqeq2d 2835 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵)) |
17 | 13, 15, 16 | cbvrex 3380 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵) |
18 | 17 | abbii 2944 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵} |
19 | 12, 18 | syl6eleqr 2917 | 1 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ⊤wtru 1657 ∈ wcel 2164 {cab 2811 ∃wrex 3118 Vcvv 3414 ⦋csb 3757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-v 3416 df-sbc 3663 df-csb 3758 |
This theorem is referenced by: eusvobj2 6903 lss1d 19329 prdsxmetlem 22550 prdsbl 22673 itg2monolem1 23923 heibor1 34150 dihglblem5 37372 |
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