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Mirrors > Home > MPE Home > Th. List > fliftval | Structured version Visualization version GIF version |
Description: The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
flift.1 | ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) |
flift.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) |
flift.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) |
fliftval.4 | ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐶) |
fliftval.5 | ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷) |
fliftval.6 | ⊢ (𝜑 → Fun 𝐹) |
Ref | Expression |
---|---|
fliftval | ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → (𝐹‘𝐶) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fliftval.6 | . . 3 ⊢ (𝜑 → Fun 𝐹) | |
2 | 1 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → Fun 𝐹) |
3 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → 𝑌 ∈ 𝑋) | |
4 | eqidd 2727 | . . . . 5 ⊢ (𝜑 → 𝐷 = 𝐷) | |
5 | eqidd 2727 | . . . . 5 ⊢ (𝑌 ∈ 𝑋 → 𝐶 = 𝐶) | |
6 | 4, 5 | anim12ci 613 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → (𝐶 = 𝐶 ∧ 𝐷 = 𝐷)) |
7 | fliftval.4 | . . . . . . 7 ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐶) | |
8 | 7 | eqeq2d 2737 | . . . . . 6 ⊢ (𝑥 = 𝑌 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐶)) |
9 | fliftval.5 | . . . . . . 7 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷) | |
10 | 9 | eqeq2d 2737 | . . . . . 6 ⊢ (𝑥 = 𝑌 → (𝐷 = 𝐵 ↔ 𝐷 = 𝐷)) |
11 | 8, 10 | anbi12d 630 | . . . . 5 ⊢ (𝑥 = 𝑌 → ((𝐶 = 𝐴 ∧ 𝐷 = 𝐵) ↔ (𝐶 = 𝐶 ∧ 𝐷 = 𝐷))) |
12 | 11 | rspcev 3606 | . . . 4 ⊢ ((𝑌 ∈ 𝑋 ∧ (𝐶 = 𝐶 ∧ 𝐷 = 𝐷)) → ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)) |
13 | 3, 6, 12 | syl2anc 583 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)) |
14 | flift.1 | . . . . 5 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) | |
15 | flift.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) | |
16 | flift.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) | |
17 | 14, 15, 16 | fliftel 7302 | . . . 4 ⊢ (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
18 | 17 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
19 | 13, 18 | mpbird 257 | . 2 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → 𝐶𝐹𝐷) |
20 | funbrfv 6936 | . 2 ⊢ (Fun 𝐹 → (𝐶𝐹𝐷 → (𝐹‘𝐶) = 𝐷)) | |
21 | 2, 19, 20 | sylc 65 | 1 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → (𝐹‘𝐶) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 ⟨cop 4629 class class class wbr 5141 ↦ cmpt 5224 ran crn 5670 Fun wfun 6531 ‘cfv 6537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6489 df-fun 6539 df-fv 6545 |
This theorem is referenced by: qliftval 8802 cygznlem2 21463 pi1xfrval 24936 pi1coval 24942 |
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