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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1520 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj1500 35061. 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 34814 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐶 → ∀𝑓 𝑤 ∈ 𝐶) |
4 | 3 | nfcii 2892 | . . . . . 6 ⊢ Ⅎ𝑓𝐶 |
5 | 4 | nfuni 4919 | . . . . 5 ⊢ Ⅎ𝑓∪ 𝐶 |
6 | 1, 5 | nfcxfr 2901 | . . . 4 ⊢ Ⅎ𝑓𝐹 |
7 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑓𝑥 | |
8 | 6, 7 | nffv 6917 | . . 3 ⊢ Ⅎ𝑓(𝐹‘𝑥) |
9 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑓𝐺 | |
10 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑓 pred(𝑥, 𝐴, 𝑅) | |
11 | 6, 10 | nfres 6002 | . . . . 5 ⊢ Ⅎ𝑓(𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) |
12 | 7, 11 | nfop 4894 | . . . 4 ⊢ Ⅎ𝑓〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
13 | 9, 12 | nffv 6917 | . . 3 ⊢ Ⅎ𝑓(𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) |
14 | 8, 13 | nfeq 2917 | . 2 ⊢ Ⅎ𝑓(𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) |
15 | 14 | nf5ri 2193 | 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 395 ∀wal 1535 = wceq 1537 {cab 2712 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 〈cop 4637 ∪ cuni 4912 ↾ cres 5691 Fn wfn 6558 ‘cfv 6563 predc-bnj14 34681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-res 5701 df-iota 6516 df-fv 6571 |
This theorem is referenced by: bnj1501 35060 |
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