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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 3900 | . . . 4 ⊢ (𝐶𝐹𝐷 ↔ 〈𝐶, 𝐷〉 ∈ 𝐹) | |
2 | flift.1 | . . . . 5 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) | |
3 | 2 | eleq2i 2184 | . . . 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 4120 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 〈𝐴, 𝐵〉 ∈ V) | |
8 | 5, 6, 7 | syl2anc 408 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ V) |
9 | 8 | ralrimiva 2482 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 〈𝐴, 𝐵〉 ∈ V) |
10 | eqid 2117 | . . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) = (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) | |
11 | 10 | elrnmptg 4761 | . . . 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 4131 | . . . 4 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉 ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) | |
15 | 5, 6, 14 | syl2anc 408 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉 ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
16 | 15 | rexbidva 2411 | . 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 1316 ∈ wcel 1465 ∀wral 2393 ∃wrex 2394 Vcvv 2660 〈cop 3500 class class class wbr 3899 ↦ cmpt 3959 ran crn 4510 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-mpt 3961 df-cnv 4517 df-dm 4519 df-rn 4520 |
This theorem is referenced by: fliftcnv 5664 fliftfun 5665 fliftf 5668 fliftval 5669 qliftel 6477 |
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