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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 33677. 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 33436 | . . . . 5 ⊢ (𝑔 ∈ 𝐶 → ∀𝑓 𝑔 ∈ 𝐶) |
3 | 2 | nf5i 2143 | . . . 4 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐶 |
4 | nfv 1918 | . . . 4 ⊢ Ⅎ𝑓Fun 𝑔 | |
5 | 3, 4 | nfim 1900 | . . 3 ⊢ Ⅎ𝑓(𝑔 ∈ 𝐶 → Fun 𝑔) |
6 | eleq1w 2821 | . . . 4 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶)) | |
7 | funeq 6522 | . . . 4 ⊢ (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔)) | |
8 | 6, 7 | imbi12d 345 | . . 3 ⊢ (𝑓 = 𝑔 → ((𝑓 ∈ 𝐶 → Fun 𝑓) ↔ (𝑔 ∈ 𝐶 → Fun 𝑔))) |
9 | 1 | bnj1436 33454 | . . . . . 6 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))) |
10 | 9 | bnj1299 33433 | . . . . 5 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑓 Fn 𝑑) |
11 | fnfun 6603 | . . . . 5 ⊢ (𝑓 Fn 𝑑 → Fun 𝑓) | |
12 | 10, 11 | bnj31 33334 | . . . 4 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 Fun 𝑓) |
13 | 12 | bnj1265 33427 | . . 3 ⊢ (𝑓 ∈ 𝐶 → Fun 𝑓) |
14 | 5, 8, 13 | chvarfv 2234 | . 2 ⊢ (𝑔 ∈ 𝐶 → Fun 𝑔) |
15 | 14 | rgen 3067 | 1 ⊢ ∀𝑔 ∈ 𝐶 Fun 𝑔 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2714 ∀wral 3065 ∃wrex 3074 ⊆ wss 3911 ⟨cop 4593 ↾ cres 5636 Fun wfun 6491 Fn wfn 6492 ‘cfv 6497 predc-bnj14 33303 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-v 3448 df-in 3918 df-ss 3928 df-br 5107 df-opab 5169 df-rel 5641 df-cnv 5642 df-co 5643 df-fun 6499 df-fn 6500 |
This theorem is referenced by: bnj60 33677 |
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