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Mirrors > Home > ILE Home > Th. List > fliftval | 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 276 | . 2 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → Fun 𝐹) |
3 | simpr 110 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → 𝑌 ∈ 𝑋) | |
4 | eqidd 2178 | . . . . 5 ⊢ (𝜑 → 𝐷 = 𝐷) | |
5 | eqidd 2178 | . . . . 5 ⊢ (𝑌 ∈ 𝑋 → 𝐶 = 𝐶) | |
6 | 4, 5 | anim12ci 339 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → (𝐶 = 𝐶 ∧ 𝐷 = 𝐷)) |
7 | fliftval.4 | . . . . . . 7 ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐶) | |
8 | 7 | eqeq2d 2189 | . . . . . 6 ⊢ (𝑥 = 𝑌 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐶)) |
9 | fliftval.5 | . . . . . . 7 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷) | |
10 | 9 | eqeq2d 2189 | . . . . . 6 ⊢ (𝑥 = 𝑌 → (𝐷 = 𝐵 ↔ 𝐷 = 𝐷)) |
11 | 8, 10 | anbi12d 473 | . . . . 5 ⊢ (𝑥 = 𝑌 → ((𝐶 = 𝐴 ∧ 𝐷 = 𝐵) ↔ (𝐶 = 𝐶 ∧ 𝐷 = 𝐷))) |
12 | 11 | rspcev 2841 | . . . 4 ⊢ ((𝑌 ∈ 𝑋 ∧ (𝐶 = 𝐶 ∧ 𝐷 = 𝐷)) → ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)) |
13 | 3, 6, 12 | syl2anc 411 | . . 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 5788 | . . . 4 ⊢ (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
18 | 17 | adantr 276 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
19 | 13, 18 | mpbird 167 | . 2 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → 𝐶𝐹𝐷) |
20 | funbrfv 5550 | . 2 ⊢ (Fun 𝐹 → (𝐶𝐹𝐷 → (𝐹‘𝐶) = 𝐷)) | |
21 | 2, 19, 20 | sylc 62 | 1 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → (𝐹‘𝐶) = 𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 〈cop 3594 class class class wbr 4000 ↦ cmpt 4061 ran crn 4624 Fun wfun 5206 ‘cfv 5212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-iota 5174 df-fun 5214 df-fv 5220 |
This theorem is referenced by: qliftval 6615 |
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