![]() |
Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1497 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 34073. 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 33832 | . . . . 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 2817 | . . . 4 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶)) | |
7 | funeq 6569 | . . . 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 33850 | . . . . . 6 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))) |
10 | 9 | bnj1299 33829 | . . . . 5 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑓 Fn 𝑑) |
11 | fnfun 6650 | . . . . 5 ⊢ (𝑓 Fn 𝑑 → Fun 𝑓) | |
12 | 10, 11 | bnj31 33730 | . . . 4 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 Fun 𝑓) |
13 | 12 | bnj1265 33823 | . . 3 ⊢ (𝑓 ∈ 𝐶 → Fun 𝑓) |
14 | 5, 8, 13 | chvarfv 2234 | . 2 ⊢ (𝑔 ∈ 𝐶 → Fun 𝑔) |
15 | 14 | rgen 3064 | 1 ⊢ ∀𝑔 ∈ 𝐶 Fun 𝑔 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2710 ∀wral 3062 ∃wrex 3071 ⊆ wss 3949 ⟨cop 4635 ↾ cres 5679 Fun wfun 6538 Fn wfn 6539 ‘cfv 6544 predc-bnj14 33699 |
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 2704 |
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 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-v 3477 df-in 3956 df-ss 3966 df-br 5150 df-opab 5212 df-rel 5684 df-cnv 5685 df-co 5686 df-fun 6546 df-fn 6547 |
This theorem is referenced by: bnj60 34073 |
Copyright terms: Public domain | W3C validator |