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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 3812 | . . . 4 ⊢ (𝐶𝐹𝐷 ↔ 〈𝐶, 𝐷〉 ∈ 𝐹) | |
2 | flift.1 | . . . . 5 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) | |
3 | 2 | eleq2i 2149 | . . . 4 ⊢ (〈𝐶, 𝐷〉 ∈ 𝐹 ↔ 〈𝐶, 𝐷〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)) |
4 | 1, 3 | bitri 182 | . . 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 4019 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 〈𝐴, 𝐵〉 ∈ V) | |
8 | 5, 6, 7 | syl2anc 403 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ V) |
9 | 8 | ralrimiva 2440 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 〈𝐴, 𝐵〉 ∈ V) |
10 | eqid 2083 | . . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) = (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) | |
11 | 10 | elrnmptg 4645 | . . . 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 190 | . 2 ⊢ (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉)) |
14 | opthg2 4030 | . . . 4 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉 ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) | |
15 | 5, 6, 14 | syl2anc 403 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉 ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
16 | 15 | rexbidva 2371 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝑋 〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
17 | 13, 16 | bitrd 186 | 1 ⊢ (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1285 ∈ wcel 1434 ∀wral 2353 ∃wrex 2354 Vcvv 2612 〈cop 3425 class class class wbr 3811 ↦ cmpt 3865 ran crn 4402 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-br 3812 df-opab 3866 df-mpt 3867 df-cnv 4409 df-dm 4411 df-rn 4412 |
This theorem is referenced by: fliftcnv 5514 fliftfun 5515 fliftf 5518 fliftval 5519 qliftel 6302 |
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