Theorem bnj1097 31656

Description: Technical lemma for bnj69 31685. 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.)

Hypotheses

Ref	Expression
bnj1097.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1097.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj1097.5	⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))

Assertion

Ref	Expression
bnj1097	⊢ ((𝑖 = ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓‘𝑖) ⊆ 𝐵)

Proof of Theorem bnj1097

Step	Hyp	Ref	Expression
1		bnj1097.3	. . . . . . . 8 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
2		bnj1097.1	. . . . . . . . 9 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
3	2	biimpi 208	. . . . . . . 8 ⊢ (𝜑 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
4	1, 3	bnj771 31441	. . . . . . 7 ⊢ (𝜒 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
5	4	3ad2ant3 1126	. . . . . 6 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
6	5	adantr 474	. . . . 5 ⊢ (((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
7		bnj1097.5	. . . . . . . 8 ⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
8	7	simp3bi 1138	. . . . . . 7 ⊢ (𝜏 → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
9	8	3ad2ant2 1125	. . . . . 6 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
10	9	adantr 474	. . . . 5 ⊢ (((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
11	6, 10	jca 507	. . . 4 ⊢ (((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
12	11	anim2i 610	. . 3 ⊢ ((𝑖 = ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑖 = ∅ ∧ ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)))
13		3anass 1079	. . 3 ⊢ ((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) ↔ (𝑖 = ∅ ∧ ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)))
14	12, 13	sylibr 226	. 2 ⊢ ((𝑖 = ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
15		fveqeq2 6457	. . . . . 6 ⊢ (𝑖 = ∅ → ((𝑓‘𝑖) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)))
16	15	biimpar 471	. . . . 5 ⊢ ((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) → (𝑓‘𝑖) = pred(𝑋, 𝐴, 𝑅))
17	16	adantr 474	. . . 4 ⊢ (((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓‘𝑖) = pred(𝑋, 𝐴, 𝑅))
18		simpr 479	. . . 4 ⊢ (((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
19	17, 18	eqsstrd 3858	. . 3 ⊢ (((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓‘𝑖) ⊆ 𝐵)
20	19	3impa 1097	. 2 ⊢ ((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓‘𝑖) ⊆ 𝐵)
21	14, 20	syl 17	1 ⊢ ((𝑖 = ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓‘𝑖) ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  Vcvv 3398   ⊆ wss 3792  ∅c0 4141   Fn wfn 6132  ‘cfv 6137   ∧ w-bnj17 31362   predc-bnj14 31364   TrFow-bnj19 31372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-iota 6101  df-fv 6145  df-bnj17 31363

This theorem is referenced by:  bnj1030  31662