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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1234 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 35055. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1234.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1234.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1234.4 | ⊢ 𝑍 = 〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1234.5 | ⊢ 𝐷 = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍))} |
Ref | Expression |
---|---|
bnj1234 | ⊢ 𝐶 = 𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fneq1 6660 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑑 ↔ 𝑔 Fn 𝑑)) | |
2 | fveq1 6906 | . . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑥) = (𝑔‘𝑥)) | |
3 | reseq1 5994 | . . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))) | |
4 | 3 | opeq2d 4885 | . . . . . . . . 9 ⊢ (𝑓 = 𝑔 → 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 = 〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉) |
5 | bnj1234.2 | . . . . . . . . 9 ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
6 | bnj1234.4 | . . . . . . . . 9 ⊢ 𝑍 = 〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
7 | 4, 5, 6 | 3eqtr4g 2800 | . . . . . . . 8 ⊢ (𝑓 = 𝑔 → 𝑌 = 𝑍) |
8 | 7 | fveq2d 6911 | . . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑌) = (𝐺‘𝑍)) |
9 | 2, 8 | eqeq12d 2751 | . . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑥) = (𝐺‘𝑌) ↔ (𝑔‘𝑥) = (𝐺‘𝑍))) |
10 | 9 | ralbidv 3176 | . . . . 5 ⊢ (𝑓 = 𝑔 → (∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌) ↔ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍))) |
11 | 1, 10 | anbi12d 632 | . . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) ↔ (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍)))) |
12 | 11 | rexbidv 3177 | . . 3 ⊢ (𝑓 = 𝑔 → (∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) ↔ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍)))) |
13 | 12 | cbvabv 2810 | . 2 ⊢ {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍))} |
14 | bnj1234.3 | . 2 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
15 | bnj1234.5 | . 2 ⊢ 𝐷 = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍))} | |
16 | 13, 14, 15 | 3eqtr4i 2773 | 1 ⊢ 𝐶 = 𝐷 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 {cab 2712 ∀wral 3059 ∃wrex 3068 〈cop 4637 ↾ cres 5691 Fn wfn 6558 ‘cfv 6563 predc-bnj14 34681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-res 5701 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 |
This theorem is referenced by: bnj1245 35007 bnj1256 35008 bnj1259 35009 bnj1296 35014 bnj1311 35017 |
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