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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1520 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj1500 33048. 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 |
---|---|
bnj1520.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1520.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1520.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1520.4 | ⊢ 𝐹 = ∪ 𝐶 |
Ref | Expression |
---|---|
bnj1520 | ⊢ ((𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) → ∀𝑓(𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1520.4 | . . . . 5 ⊢ 𝐹 = ∪ 𝐶 | |
2 | bnj1520.3 | . . . . . . . 8 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
3 | 2 | bnj1317 32801 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐶 → ∀𝑓 𝑤 ∈ 𝐶) |
4 | 3 | nfcii 2891 | . . . . . 6 ⊢ Ⅎ𝑓𝐶 |
5 | 4 | nfuni 4846 | . . . . 5 ⊢ Ⅎ𝑓∪ 𝐶 |
6 | 1, 5 | nfcxfr 2905 | . . . 4 ⊢ Ⅎ𝑓𝐹 |
7 | nfcv 2907 | . . . 4 ⊢ Ⅎ𝑓𝑥 | |
8 | 6, 7 | nffv 6784 | . . 3 ⊢ Ⅎ𝑓(𝐹‘𝑥) |
9 | nfcv 2907 | . . . 4 ⊢ Ⅎ𝑓𝐺 | |
10 | nfcv 2907 | . . . . . 6 ⊢ Ⅎ𝑓 pred(𝑥, 𝐴, 𝑅) | |
11 | 6, 10 | nfres 5893 | . . . . 5 ⊢ Ⅎ𝑓(𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) |
12 | 7, 11 | nfop 4820 | . . . 4 ⊢ Ⅎ𝑓〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
13 | 9, 12 | nffv 6784 | . . 3 ⊢ Ⅎ𝑓(𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) |
14 | 8, 13 | nfeq 2920 | . 2 ⊢ Ⅎ𝑓(𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) |
15 | 14 | nf5ri 2188 | 1 ⊢ ((𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) → ∀𝑓(𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1537 = wceq 1539 {cab 2715 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 〈cop 4567 ∪ cuni 4839 ↾ cres 5591 Fn wfn 6428 ‘cfv 6433 predc-bnj14 32667 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-xp 5595 df-res 5601 df-iota 6391 df-fv 6441 |
This theorem is referenced by: bnj1501 33047 |
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