Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1497 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 32709. 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 |
---|---|
bnj1497.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1497.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1497.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
Ref | Expression |
---|---|
bnj1497 | ⊢ ∀𝑔 ∈ 𝐶 Fun 𝑔 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1497.3 | . . . . . 6 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
2 | 1 | bnj1317 32468 | . . . . 5 ⊢ (𝑔 ∈ 𝐶 → ∀𝑓 𝑔 ∈ 𝐶) |
3 | 2 | nf5i 2148 | . . . 4 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐶 |
4 | nfv 1922 | . . . 4 ⊢ Ⅎ𝑓Fun 𝑔 | |
5 | 3, 4 | nfim 1904 | . . 3 ⊢ Ⅎ𝑓(𝑔 ∈ 𝐶 → Fun 𝑔) |
6 | eleq1w 2813 | . . . 4 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶)) | |
7 | funeq 6378 | . . . 4 ⊢ (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔)) | |
8 | 6, 7 | imbi12d 348 | . . 3 ⊢ (𝑓 = 𝑔 → ((𝑓 ∈ 𝐶 → Fun 𝑓) ↔ (𝑔 ∈ 𝐶 → Fun 𝑔))) |
9 | 1 | bnj1436 32486 | . . . . . 6 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))) |
10 | 9 | bnj1299 32465 | . . . . 5 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑓 Fn 𝑑) |
11 | fnfun 6457 | . . . . 5 ⊢ (𝑓 Fn 𝑑 → Fun 𝑓) | |
12 | 10, 11 | bnj31 32364 | . . . 4 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 Fun 𝑓) |
13 | 12 | bnj1265 32459 | . . 3 ⊢ (𝑓 ∈ 𝐶 → Fun 𝑓) |
14 | 5, 8, 13 | chvarfv 2240 | . 2 ⊢ (𝑔 ∈ 𝐶 → Fun 𝑔) |
15 | 14 | rgen 3061 | 1 ⊢ ∀𝑔 ∈ 𝐶 Fun 𝑔 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 {cab 2714 ∀wral 3051 ∃wrex 3052 ⊆ wss 3853 〈cop 4533 ↾ cres 5538 Fun wfun 6352 Fn wfn 6353 ‘cfv 6358 predc-bnj14 32333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-v 3400 df-in 3860 df-ss 3870 df-br 5040 df-opab 5102 df-rel 5543 df-cnv 5544 df-co 5545 df-fun 6360 df-fn 6361 |
This theorem is referenced by: bnj60 32709 |
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