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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 31458. 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 6140 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑑 ↔ 𝑔 Fn 𝑑)) | |
2 | fveq1 6352 | . . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑥) = (𝑔‘𝑥)) | |
3 | reseq1 5545 | . . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))) | |
4 | 3 | opeq2d 4560 | . . . . . . . . 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 2819 | . . . . . . . 8 ⊢ (𝑓 = 𝑔 → 𝑌 = 𝑍) |
8 | 7 | fveq2d 6357 | . . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑌) = (𝐺‘𝑍)) |
9 | 2, 8 | eqeq12d 2775 | . . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑥) = (𝐺‘𝑌) ↔ (𝑔‘𝑥) = (𝐺‘𝑍))) |
10 | 9 | ralbidv 3124 | . . . . 5 ⊢ (𝑓 = 𝑔 → (∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌) ↔ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍))) |
11 | 1, 10 | anbi12d 749 | . . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) ↔ (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍)))) |
12 | 11 | rexbidv 3190 | . . 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 2885 | . 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 2792 | 1 ⊢ 𝐶 = 𝐷 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1632 {cab 2746 ∀wral 3050 ∃wrex 3051 〈cop 4327 ↾ cres 5268 Fn wfn 6044 ‘cfv 6049 predc-bnj14 31084 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-res 5278 df-iota 6012 df-fun 6051 df-fn 6052 df-fv 6057 |
This theorem is referenced by: bnj1245 31410 bnj1256 31411 bnj1259 31412 bnj1296 31417 bnj1311 31420 |
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