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Mirrors > Home > MPE Home > Th. List > kqfval | Structured version Visualization version GIF version |
Description: Value of the function appearing in df-kq 22304. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
kqval.2 | ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) |
Ref | Expression |
---|---|
kqfval | ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) = {𝑦 ∈ 𝐽 ∣ 𝐴 ∈ 𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ 𝑋) | |
2 | rabexg 5236 | . 2 ⊢ (𝐽 ∈ 𝑉 → {𝑦 ∈ 𝐽 ∣ 𝐴 ∈ 𝑦} ∈ V) | |
3 | eleq1 2902 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦)) | |
4 | 3 | rabbidv 3482 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} = {𝑦 ∈ 𝐽 ∣ 𝐴 ∈ 𝑦}) |
5 | kqval.2 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) | |
6 | 4, 5 | fvmptg 6768 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ {𝑦 ∈ 𝐽 ∣ 𝐴 ∈ 𝑦} ∈ V) → (𝐹‘𝐴) = {𝑦 ∈ 𝐽 ∣ 𝐴 ∈ 𝑦}) |
7 | 1, 2, 6 | syl2anr 598 | 1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) = {𝑦 ∈ 𝐽 ∣ 𝐴 ∈ 𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3144 Vcvv 3496 ↦ cmpt 5148 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 |
This theorem is referenced by: kqfeq 22334 isr0 22347 |
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