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Mirrors > Home > MPE Home > Th. List > elqtop | Structured version Visualization version GIF version |
Description: Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.) |
Ref | Expression |
---|---|
qtopval.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
elqtop | ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐴 ∈ (𝐽 qTop 𝐹) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ 𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qtopval.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | qtopval2 22301 | . . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 𝑌 ∣ (◡𝐹 “ 𝑠) ∈ 𝐽}) |
3 | 2 | eleq2d 2875 | . 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐴 ∈ (𝐽 qTop 𝐹) ↔ 𝐴 ∈ {𝑠 ∈ 𝒫 𝑌 ∣ (◡𝐹 “ 𝑠) ∈ 𝐽})) |
4 | imaeq2 5892 | . . . . 5 ⊢ (𝑠 = 𝐴 → (◡𝐹 “ 𝑠) = (◡𝐹 “ 𝐴)) | |
5 | 4 | eleq1d 2874 | . . . 4 ⊢ (𝑠 = 𝐴 → ((◡𝐹 “ 𝑠) ∈ 𝐽 ↔ (◡𝐹 “ 𝐴) ∈ 𝐽)) |
6 | 5 | elrab 3628 | . . 3 ⊢ (𝐴 ∈ {𝑠 ∈ 𝒫 𝑌 ∣ (◡𝐹 “ 𝑠) ∈ 𝐽} ↔ (𝐴 ∈ 𝒫 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ 𝐽)) |
7 | uniexg 7446 | . . . . . . . . 9 ⊢ (𝐽 ∈ 𝑉 → ∪ 𝐽 ∈ V) | |
8 | 1, 7 | eqeltrid 2894 | . . . . . . . 8 ⊢ (𝐽 ∈ 𝑉 → 𝑋 ∈ V) |
9 | 8 | 3ad2ant1 1130 | . . . . . . 7 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑋 ∈ V) |
10 | simp3 1135 | . . . . . . 7 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑍 ⊆ 𝑋) | |
11 | 9, 10 | ssexd 5192 | . . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑍 ∈ V) |
12 | simp2 1134 | . . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝐹:𝑍–onto→𝑌) | |
13 | fornex 7639 | . . . . . 6 ⊢ (𝑍 ∈ V → (𝐹:𝑍–onto→𝑌 → 𝑌 ∈ V)) | |
14 | 11, 12, 13 | sylc 65 | . . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑌 ∈ V) |
15 | elpw2g 5211 | . . . . 5 ⊢ (𝑌 ∈ V → (𝐴 ∈ 𝒫 𝑌 ↔ 𝐴 ⊆ 𝑌)) | |
16 | 14, 15 | syl 17 | . . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐴 ∈ 𝒫 𝑌 ↔ 𝐴 ⊆ 𝑌)) |
17 | 16 | anbi1d 632 | . . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → ((𝐴 ∈ 𝒫 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ 𝐽) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ 𝐽))) |
18 | 6, 17 | syl5bb 286 | . 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐴 ∈ {𝑠 ∈ 𝒫 𝑌 ∣ (◡𝐹 “ 𝑠) ∈ 𝐽} ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ 𝐽))) |
19 | 3, 18 | bitrd 282 | 1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐴 ∈ (𝐽 qTop 𝐹) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ 𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ⊆ wss 3881 𝒫 cpw 4497 ∪ cuni 4800 ◡ccnv 5518 “ cima 5522 –onto→wfo 6322 (class class class)co 7135 qTop cqtop 16768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-qtop 16772 |
This theorem is referenced by: qtoptop2 22304 elqtop2 22306 elqtop3 22308 |
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