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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1497 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj60 35076. 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 34835 | . . . . 5 ⊢ (𝑔 ∈ 𝐶 → ∀𝑓 𝑔 ∈ 𝐶) |
| 3 | 2 | nf5i 2146 | . . . 4 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐶 |
| 4 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑓Fun 𝑔 | |
| 5 | 3, 4 | nfim 1896 | . . 3 ⊢ Ⅎ𝑓(𝑔 ∈ 𝐶 → Fun 𝑔) |
| 6 | eleq1w 2824 | . . . 4 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶)) | |
| 7 | funeq 6586 | . . . 4 ⊢ (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔)) | |
| 8 | 6, 7 | imbi12d 344 | . . 3 ⊢ (𝑓 = 𝑔 → ((𝑓 ∈ 𝐶 → Fun 𝑓) ↔ (𝑔 ∈ 𝐶 → Fun 𝑔))) |
| 9 | 1 | bnj1436 34853 | . . . . . 6 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))) |
| 10 | 9 | bnj1299 34832 | . . . . 5 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑓 Fn 𝑑) |
| 11 | fnfun 6668 | . . . . 5 ⊢ (𝑓 Fn 𝑑 → Fun 𝑓) | |
| 12 | 10, 11 | bnj31 34733 | . . . 4 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 Fun 𝑓) |
| 13 | 12 | bnj1265 34826 | . . 3 ⊢ (𝑓 ∈ 𝐶 → Fun 𝑓) |
| 14 | 5, 8, 13 | chvarfv 2240 | . 2 ⊢ (𝑔 ∈ 𝐶 → Fun 𝑔) |
| 15 | 14 | rgen 3063 | 1 ⊢ ∀𝑔 ∈ 𝐶 Fun 𝑔 |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 〈cop 4632 ↾ cres 5687 Fun wfun 6555 Fn wfn 6556 ‘cfv 6561 predc-bnj14 34702 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-ss 3968 df-br 5144 df-opab 5206 df-rel 5692 df-cnv 5693 df-co 5694 df-fun 6563 df-fn 6564 |
| This theorem is referenced by: bnj60 35076 |
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