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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdordlem1 | Structured version Visualization version GIF version |
Description: Lemma for mapdord 40813. (Contributed by NM, 27-Jan-2015.) |
Ref | Expression |
---|---|
mapdordlem1.t | ⊢ 𝑇 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) ∈ 𝑌} |
Ref | Expression |
---|---|
mapdordlem1 | ⊢ (𝐽 ∈ 𝑇 ↔ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2fveq3 6897 | . . . 4 ⊢ (𝑔 = 𝐽 → (𝑂‘(𝐿‘𝑔)) = (𝑂‘(𝐿‘𝐽))) | |
2 | 1 | fveq2d 6896 | . . 3 ⊢ (𝑔 = 𝐽 → (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝑂‘(𝑂‘(𝐿‘𝐽)))) |
3 | 2 | eleq1d 2817 | . 2 ⊢ (𝑔 = 𝐽 → ((𝑂‘(𝑂‘(𝐿‘𝑔))) ∈ 𝑌 ↔ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)) |
4 | mapdordlem1.t | . 2 ⊢ 𝑇 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) ∈ 𝑌} | |
5 | 3, 4 | elrab2 3687 | 1 ⊢ (𝐽 ∈ 𝑇 ↔ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 {crab 3431 ‘cfv 6544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 |
This theorem is referenced by: mapdordlem2 40812 |
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