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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 3977 | . . . 4 ⊢ (𝐶𝐹𝐷 ↔ 〈𝐶, 𝐷〉 ∈ 𝐹) | |
2 | flift.1 | . . . . 5 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) | |
3 | 2 | eleq2i 2231 | . . . 4 ⊢ (〈𝐶, 𝐷〉 ∈ 𝐹 ↔ 〈𝐶, 𝐷〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)) |
4 | 1, 3 | bitri 183 | . . 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 4200 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 〈𝐴, 𝐵〉 ∈ V) | |
8 | 5, 6, 7 | syl2anc 409 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ V) |
9 | 8 | ralrimiva 2537 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 〈𝐴, 𝐵〉 ∈ V) |
10 | eqid 2164 | . . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) = (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) | |
11 | 10 | elrnmptg 4850 | . . . 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 | syl5bb 191 | . 2 ⊢ (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉)) |
14 | opthg2 4211 | . . . 4 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉 ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) | |
15 | 5, 6, 14 | syl2anc 409 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉 ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
16 | 15 | rexbidva 2461 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝑋 〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
17 | 13, 16 | bitrd 187 | 1 ⊢ (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 ∀wral 2442 ∃wrex 2443 Vcvv 2721 〈cop 3573 class class class wbr 3976 ↦ cmpt 4037 ran crn 4599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-mpt 4039 df-cnv 4606 df-dm 4608 df-rn 4609 |
This theorem is referenced by: fliftcnv 5757 fliftfun 5758 fliftf 5761 fliftval 5762 qliftel 6572 |
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