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 32231. 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 31992 | . . . . 5 ⊢ (𝑔 ∈ 𝐶 → ∀𝑓 𝑔 ∈ 𝐶) |
3 | 2 | nf5i 2141 | . . . 4 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐶 |
4 | nfv 1906 | . . . 4 ⊢ Ⅎ𝑓Fun 𝑔 | |
5 | 3, 4 | nfim 1888 | . . 3 ⊢ Ⅎ𝑓(𝑔 ∈ 𝐶 → Fun 𝑔) |
6 | eleq1w 2892 | . . . 4 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶)) | |
7 | funeq 6368 | . . . 4 ⊢ (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔)) | |
8 | 6, 7 | imbi12d 346 | . . 3 ⊢ (𝑓 = 𝑔 → ((𝑓 ∈ 𝐶 → Fun 𝑓) ↔ (𝑔 ∈ 𝐶 → Fun 𝑔))) |
9 | 1 | bnj1436 32010 | . . . . . 6 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))) |
10 | 9 | bnj1299 31989 | . . . . 5 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑓 Fn 𝑑) |
11 | fnfun 6446 | . . . . 5 ⊢ (𝑓 Fn 𝑑 → Fun 𝑓) | |
12 | 10, 11 | bnj31 31888 | . . . 4 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 Fun 𝑓) |
13 | 12 | bnj1265 31983 | . . 3 ⊢ (𝑓 ∈ 𝐶 → Fun 𝑓) |
14 | 5, 8, 13 | chvarfv 2232 | . 2 ⊢ (𝑔 ∈ 𝐶 → Fun 𝑔) |
15 | 14 | rgen 3145 | 1 ⊢ ∀𝑔 ∈ 𝐶 Fun 𝑔 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {cab 2796 ∀wral 3135 ∃wrex 3136 ⊆ wss 3933 〈cop 4563 ↾ cres 5550 Fun wfun 6342 Fn wfn 6343 ‘cfv 6348 predc-bnj14 31857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-in 3940 df-ss 3949 df-br 5058 df-opab 5120 df-rel 5555 df-cnv 5556 df-co 5557 df-fun 6350 df-fn 6351 |
This theorem is referenced by: bnj60 32231 |
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