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Mirrors > Home > ILE Home > Th. List > fliftel | GIF version |
Description: Elementhood in the relation 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) |
Ref | Expression |
---|---|
fliftel | ⊢ (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 4001 | . . . 4 ⊢ (𝐶𝐹𝐷 ↔ 〈𝐶, 𝐷〉 ∈ 𝐹) | |
2 | flift.1 | . . . . 5 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) | |
3 | 2 | eleq2i 2244 | . . . 4 ⊢ (〈𝐶, 𝐷〉 ∈ 𝐹 ↔ 〈𝐶, 𝐷〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)) |
4 | 1, 3 | bitri 184 | . . 3 ⊢ (𝐶𝐹𝐷 ↔ 〈𝐶, 𝐷〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)) |
5 | flift.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) | |
6 | flift.3 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) | |
7 | opexg 4225 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 〈𝐴, 𝐵〉 ∈ V) | |
8 | 5, 6, 7 | syl2anc 411 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ V) |
9 | 8 | ralrimiva 2550 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 〈𝐴, 𝐵〉 ∈ V) |
10 | eqid 2177 | . . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) = (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) | |
11 | 10 | elrnmptg 4875 | . . . 4 ⊢ (∀𝑥 ∈ 𝑋 〈𝐴, 𝐵〉 ∈ V → (〈𝐶, 𝐷〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) ↔ ∃𝑥 ∈ 𝑋 〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉)) |
12 | 9, 11 | syl 14 | . . 3 ⊢ (𝜑 → (〈𝐶, 𝐷〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) ↔ ∃𝑥 ∈ 𝑋 〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉)) |
13 | 4, 12 | bitrid 192 | . 2 ⊢ (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉)) |
14 | opthg2 4236 | . . . 4 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉 ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) | |
15 | 5, 6, 14 | syl2anc 411 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉 ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
16 | 15 | rexbidva 2474 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝑋 〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
17 | 13, 16 | bitrd 188 | 1 ⊢ (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 Vcvv 2737 〈cop 3594 class class class wbr 4000 ↦ cmpt 4061 ran crn 4624 |
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-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-mpt 4063 df-cnv 4631 df-dm 4633 df-rn 4634 |
This theorem is referenced by: fliftcnv 5790 fliftfun 5791 fliftf 5794 fliftval 5795 qliftel 6609 |
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