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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1256 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 32613. 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 |
---|---|
bnj1256.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1256.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1256.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1256.4 | ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ) |
bnj1256.5 | ⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} |
bnj1256.6 | ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷))) |
bnj1256.7 | ⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥)) |
Ref | Expression |
---|---|
bnj1256 | ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1256.6 | . 2 ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷))) | |
2 | abid 2720 | . . . 4 ⊢ (𝑔 ∈ {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉))} ↔ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉))) | |
3 | 2 | bnj1238 32357 | . . 3 ⊢ (𝑔 ∈ {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉))} → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑) |
4 | bnj1256.2 | . . . 4 ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
5 | bnj1256.3 | . . . 4 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
6 | eqid 2738 | . . . 4 ⊢ 〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉 = 〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
7 | eqid 2738 | . . . 4 ⊢ {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉))} = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉))} | |
8 | 4, 5, 6, 7 | bnj1234 32564 | . . 3 ⊢ 𝐶 = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘〈𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))〉))} |
9 | 3, 8 | eleq2s 2851 | . 2 ⊢ (𝑔 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑) |
10 | 1, 9 | bnj770 32313 | 1 ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 {cab 2716 ≠ wne 2934 ∀wral 3053 ∃wrex 3054 {crab 3057 ∩ cin 3842 ⊆ wss 3843 〈cop 4522 class class class wbr 5030 dom cdm 5525 ↾ cres 5527 Fn wfn 6334 ‘cfv 6339 ∧ w-bnj17 32235 predc-bnj14 32237 FrSe w-bnj15 32241 |
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-12 2179 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-un 3848 df-in 3850 df-ss 3860 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-res 5537 df-iota 6297 df-fun 6341 df-fn 6342 df-fv 6347 df-bnj17 32236 |
This theorem is referenced by: bnj1253 32568 bnj1286 32570 bnj1280 32571 |
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