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Mirrors > Home > MPE Home > Th. List > fliftel | Structured version Visualization version 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 4584 | . . 3 ⊢ (𝐶𝐹𝐷 ↔ 〈𝐶, 𝐷〉 ∈ 𝐹) | |
2 | flift.1 | . . . 4 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) | |
3 | 2 | eleq2i 2680 | . . 3 ⊢ (〈𝐶, 𝐷〉 ∈ 𝐹 ↔ 〈𝐶, 𝐷〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)) |
4 | eqid 2610 | . . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) = (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) | |
5 | opex 4859 | . . . 4 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
6 | 4, 5 | elrnmpti 5297 | . . 3 ⊢ (〈𝐶, 𝐷〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) ↔ ∃𝑥 ∈ 𝑋 〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉) |
7 | 1, 3, 6 | 3bitri 285 | . 2 ⊢ (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉) |
8 | flift.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) | |
9 | flift.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) | |
10 | opthg2 4874 | . . . 4 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉 ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) | |
11 | 8, 9, 10 | syl2anc 691 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉 ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
12 | 11 | rexbidva 3031 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝑋 〈𝐶, 𝐷〉 = 〈𝐴, 𝐵〉 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
13 | 7, 12 | syl5bb 271 | 1 ⊢ (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 〈cop 4131 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-mpt 4645 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: fliftcnv 6461 fliftfun 6462 fliftf 6465 fliftval 6466 qliftel 7717 |
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