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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1259 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 31130. 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 |
---|---|
bnj1259.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1259.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1259.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1259.4 | ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ) |
bnj1259.5 | ⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} |
bnj1259.6 | ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷))) |
bnj1259.7 | ⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥)) |
Ref | Expression |
---|---|
bnj1259 | ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 ℎ Fn 𝑑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1259.6 | . 2 ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷))) | |
2 | abid 2610 | . . . 4 ⊢ (ℎ ∈ {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘〈𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))〉))} ↔ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘〈𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))〉))) | |
3 | 2 | bnj1238 30877 | . . 3 ⊢ (ℎ ∈ {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘〈𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))〉))} → ∃𝑑 ∈ 𝐵 ℎ Fn 𝑑) |
4 | bnj1259.2 | . . . 4 ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
5 | bnj1259.3 | . . . 4 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
6 | eqid 2622 | . . . 4 ⊢ 〈𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))〉 = 〈𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
7 | eqid 2622 | . . . 4 ⊢ {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘〈𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))〉))} = {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘〈𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))〉))} | |
8 | 4, 5, 6, 7 | bnj1234 31081 | . . 3 ⊢ 𝐶 = {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘〈𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))〉))} |
9 | 3, 8 | eleq2s 2719 | . 2 ⊢ (ℎ ∈ 𝐶 → ∃𝑑 ∈ 𝐵 ℎ Fn 𝑑) |
10 | 1, 9 | bnj771 30834 | 1 ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 ℎ Fn 𝑑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 {cab 2608 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 ∩ cin 3573 ⊆ wss 3574 〈cop 4183 class class class wbr 4653 dom cdm 5114 ↾ cres 5116 Fn wfn 5883 ‘cfv 5888 ∧ w-bnj17 30752 predc-bnj14 30754 FrSe w-bnj15 30758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-res 5126 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 df-bnj17 30753 |
This theorem is referenced by: bnj1253 31085 bnj1286 31087 bnj1280 31088 |
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