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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 484 | . 2 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → Fun 𝐹) |
3 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → 𝑌 ∈ 𝑋) | |
4 | eqidd 2799 | . . . . 5 ⊢ (𝜑 → 𝐷 = 𝐷) | |
5 | eqidd 2799 | . . . . 5 ⊢ (𝑌 ∈ 𝑋 → 𝐶 = 𝐶) | |
6 | 4, 5 | anim12ci 616 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → (𝐶 = 𝐶 ∧ 𝐷 = 𝐷)) |
7 | fliftval.4 | . . . . . . 7 ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐶) | |
8 | 7 | eqeq2d 2809 | . . . . . 6 ⊢ (𝑥 = 𝑌 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐶)) |
9 | fliftval.5 | . . . . . . 7 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷) | |
10 | 9 | eqeq2d 2809 | . . . . . 6 ⊢ (𝑥 = 𝑌 → (𝐷 = 𝐵 ↔ 𝐷 = 𝐷)) |
11 | 8, 10 | anbi12d 633 | . . . . 5 ⊢ (𝑥 = 𝑌 → ((𝐶 = 𝐴 ∧ 𝐷 = 𝐵) ↔ (𝐶 = 𝐶 ∧ 𝐷 = 𝐷))) |
12 | 11 | rspcev 3571 | . . . 4 ⊢ ((𝑌 ∈ 𝑋 ∧ (𝐶 = 𝐶 ∧ 𝐷 = 𝐷)) → ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)) |
13 | 3, 6, 12 | syl2anc 587 | . . 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 7041 | . . . 4 ⊢ (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
18 | 17 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
19 | 13, 18 | mpbird 260 | . 2 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → 𝐶𝐹𝐷) |
20 | funbrfv 6691 | . 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 〈cop 4531 class class class wbr 5030 ↦ cmpt 5110 ran crn 5520 Fun wfun 6318 ‘cfv 6324 |
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-sep 5167 ax-nul 5174 ax-pr 5295 |
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-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 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-iota 6283 df-fun 6326 df-fv 6332 |
This theorem is referenced by: qliftval 8369 cygznlem2 20260 pi1xfrval 23659 pi1coval 23665 |
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