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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1520 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj1500 32631. 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 32384 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐶 → ∀𝑓 𝑤 ∈ 𝐶) |
4 | 3 | nfcii 2884 | . . . . . 6 ⊢ Ⅎ𝑓𝐶 |
5 | 4 | nfuni 4813 | . . . . 5 ⊢ Ⅎ𝑓∪ 𝐶 |
6 | 1, 5 | nfcxfr 2898 | . . . 4 ⊢ Ⅎ𝑓𝐹 |
7 | nfcv 2900 | . . . 4 ⊢ Ⅎ𝑓𝑥 | |
8 | 6, 7 | nffv 6696 | . . 3 ⊢ Ⅎ𝑓(𝐹‘𝑥) |
9 | nfcv 2900 | . . . 4 ⊢ Ⅎ𝑓𝐺 | |
10 | nfcv 2900 | . . . . . 6 ⊢ Ⅎ𝑓 pred(𝑥, 𝐴, 𝑅) | |
11 | 6, 10 | nfres 5837 | . . . . 5 ⊢ Ⅎ𝑓(𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) |
12 | 7, 11 | nfop 4787 | . . . 4 ⊢ Ⅎ𝑓〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
13 | 9, 12 | nffv 6696 | . . 3 ⊢ Ⅎ𝑓(𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) |
14 | 8, 13 | nfeq 2913 | . 2 ⊢ Ⅎ𝑓(𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) |
15 | 14 | nf5ri 2197 | 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 399 ∀wal 1540 = wceq 1542 {cab 2717 ∀wral 3054 ∃wrex 3055 ⊆ wss 3853 〈cop 4532 ∪ cuni 4806 ↾ cres 5537 Fn wfn 6344 ‘cfv 6349 predc-bnj14 32249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-xp 5541 df-res 5547 df-iota 6307 df-fv 6357 |
This theorem is referenced by: bnj1501 32630 |
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