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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 21309 | . . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐽 qTop 𝐹) = {𝑠 ∈ 𝒫 𝑌 ∣ (◡𝐹 “ 𝑠) ∈ 𝐽}) |
3 | 2 | eleq2d 2673 | . 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → (𝐴 ∈ (𝐽 qTop 𝐹) ↔ 𝐴 ∈ {𝑠 ∈ 𝒫 𝑌 ∣ (◡𝐹 “ 𝑠) ∈ 𝐽})) |
4 | imaeq2 5381 | . . . . 5 ⊢ (𝑠 = 𝐴 → (◡𝐹 “ 𝑠) = (◡𝐹 “ 𝐴)) | |
5 | 4 | eleq1d 2672 | . . . 4 ⊢ (𝑠 = 𝐴 → ((◡𝐹 “ 𝑠) ∈ 𝐽 ↔ (◡𝐹 “ 𝐴) ∈ 𝐽)) |
6 | 5 | elrab 3331 | . . 3 ⊢ (𝐴 ∈ {𝑠 ∈ 𝒫 𝑌 ∣ (◡𝐹 “ 𝑠) ∈ 𝐽} ↔ (𝐴 ∈ 𝒫 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ 𝐽)) |
7 | uniexg 6853 | . . . . . . . . 9 ⊢ (𝐽 ∈ 𝑉 → ∪ 𝐽 ∈ V) | |
8 | 1, 7 | syl5eqel 2692 | . . . . . . . 8 ⊢ (𝐽 ∈ 𝑉 → 𝑋 ∈ V) |
9 | 8 | 3ad2ant1 1075 | . . . . . . 7 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑋 ∈ V) |
10 | simp3 1056 | . . . . . . 7 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑍 ⊆ 𝑋) | |
11 | 9, 10 | ssexd 4733 | . . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑍 ∈ V) |
12 | simp2 1055 | . . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝐹:𝑍–onto→𝑌) | |
13 | fornex 7028 | . . . . . 6 ⊢ (𝑍 ∈ V → (𝐹:𝑍–onto→𝑌 → 𝑌 ∈ V)) | |
14 | 11, 12, 13 | sylc 63 | . . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → 𝑌 ∈ V) |
15 | elpw2g 4754 | . . . . 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 737 | . . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐹:𝑍–onto→𝑌 ∧ 𝑍 ⊆ 𝑋) → ((𝐴 ∈ 𝒫 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ 𝐽) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ 𝐽))) |
18 | 6, 17 | syl5bb 271 | . 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 267 | 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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 ◡ccnv 5037 “ cima 5041 –onto→wfo 5802 (class class class)co 6549 qTop cqtop 15986 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-qtop 15990 |
This theorem is referenced by: qtoptop2 21312 elqtop2 21314 elqtop3 21316 |
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