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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 31940. 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 6271 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑑 ↔ 𝑔 Fn 𝑑)) | |
2 | fveq1 6492 | . . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑥) = (𝑔‘𝑥)) | |
3 | reseq1 5682 | . . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))) | |
4 | 3 | opeq2d 4678 | . . . . . . . . 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 2833 | . . . . . . . 8 ⊢ (𝑓 = 𝑔 → 𝑌 = 𝑍) |
8 | 7 | fveq2d 6497 | . . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑌) = (𝐺‘𝑍)) |
9 | 2, 8 | eqeq12d 2787 | . . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑥) = (𝐺‘𝑌) ↔ (𝑔‘𝑥) = (𝐺‘𝑍))) |
10 | 9 | ralbidv 3141 | . . . . 5 ⊢ (𝑓 = 𝑔 → (∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌) ↔ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍))) |
11 | 1, 10 | anbi12d 621 | . . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) ↔ (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘𝑍)))) |
12 | 11 | rexbidv 3236 | . . 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 2904 | . 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 2806 | 1 ⊢ 𝐶 = 𝐷 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 = wceq 1507 {cab 2753 ∀wral 3082 ∃wrex 3083 〈cop 4441 ↾ cres 5402 Fn wfn 6177 ‘cfv 6182 predc-bnj14 31567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-res 5412 df-iota 6146 df-fun 6184 df-fn 6185 df-fv 6190 |
This theorem is referenced by: bnj1245 31892 bnj1256 31893 bnj1259 31894 bnj1296 31899 bnj1311 31902 |
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