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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 479 | . 2 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → Fun 𝐹) |
3 | simpr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → 𝑌 ∈ 𝑋) | |
4 | eqidd 2729 | . . . . 5 ⊢ (𝜑 → 𝐷 = 𝐷) | |
5 | eqidd 2729 | . . . . 5 ⊢ (𝑌 ∈ 𝑋 → 𝐶 = 𝐶) | |
6 | 4, 5 | anim12ci 612 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → (𝐶 = 𝐶 ∧ 𝐷 = 𝐷)) |
7 | fliftval.4 | . . . . . . 7 ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐶) | |
8 | 7 | eqeq2d 2739 | . . . . . 6 ⊢ (𝑥 = 𝑌 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐶)) |
9 | fliftval.5 | . . . . . . 7 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷) | |
10 | 9 | eqeq2d 2739 | . . . . . 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 3611 | . . . 4 ⊢ ((𝑌 ∈ 𝑋 ∧ (𝐶 = 𝐶 ∧ 𝐷 = 𝐷)) → ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)) |
13 | 3, 6, 12 | syl2anc 582 | . . 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 7323 | . . . 4 ⊢ (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
18 | 17 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
19 | 13, 18 | mpbird 256 | . 2 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → 𝐶𝐹𝐷) |
20 | funbrfv 6953 | . 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 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3067 ⟨cop 4638 class class class wbr 5152 ↦ cmpt 5235 ran crn 5683 Fun wfun 6547 ‘cfv 6553 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fv 6561 |
This theorem is referenced by: qliftval 8833 cygznlem2 21516 pi1xfrval 25009 pi1coval 25015 |
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