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Theorem isf32lem1 10390
Description: Lemma for isfin3-2 10404. Derive weak ordering property. (Contributed by Stefan O'Rear, 5-Nov-2014.)
Hypotheses
Ref Expression
isf32lem.a (𝜑𝐹:ω⟶𝒫 𝐺)
isf32lem.b (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))
isf32lem.c (𝜑 → ¬ ran 𝐹 ∈ ran 𝐹)
Assertion
Ref Expression
isf32lem1 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝐴𝜑)) → (𝐹𝐴) ⊆ (𝐹𝐵))
Distinct variable groups:   𝑥,𝐵   𝜑,𝑥   𝑥,𝐴   𝑥,𝐹
Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem isf32lem1
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6906 . . . . 5 (𝑎 = 𝐵 → (𝐹𝑎) = (𝐹𝐵))
21sseq1d 4026 . . . 4 (𝑎 = 𝐵 → ((𝐹𝑎) ⊆ (𝐹𝐵) ↔ (𝐹𝐵) ⊆ (𝐹𝐵)))
32imbi2d 340 . . 3 (𝑎 = 𝐵 → ((𝜑 → (𝐹𝑎) ⊆ (𝐹𝐵)) ↔ (𝜑 → (𝐹𝐵) ⊆ (𝐹𝐵))))
4 fveq2 6906 . . . . 5 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
54sseq1d 4026 . . . 4 (𝑎 = 𝑏 → ((𝐹𝑎) ⊆ (𝐹𝐵) ↔ (𝐹𝑏) ⊆ (𝐹𝐵)))
65imbi2d 340 . . 3 (𝑎 = 𝑏 → ((𝜑 → (𝐹𝑎) ⊆ (𝐹𝐵)) ↔ (𝜑 → (𝐹𝑏) ⊆ (𝐹𝐵))))
7 fveq2 6906 . . . . 5 (𝑎 = suc 𝑏 → (𝐹𝑎) = (𝐹‘suc 𝑏))
87sseq1d 4026 . . . 4 (𝑎 = suc 𝑏 → ((𝐹𝑎) ⊆ (𝐹𝐵) ↔ (𝐹‘suc 𝑏) ⊆ (𝐹𝐵)))
98imbi2d 340 . . 3 (𝑎 = suc 𝑏 → ((𝜑 → (𝐹𝑎) ⊆ (𝐹𝐵)) ↔ (𝜑 → (𝐹‘suc 𝑏) ⊆ (𝐹𝐵))))
10 fveq2 6906 . . . . 5 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
1110sseq1d 4026 . . . 4 (𝑎 = 𝐴 → ((𝐹𝑎) ⊆ (𝐹𝐵) ↔ (𝐹𝐴) ⊆ (𝐹𝐵)))
1211imbi2d 340 . . 3 (𝑎 = 𝐴 → ((𝜑 → (𝐹𝑎) ⊆ (𝐹𝐵)) ↔ (𝜑 → (𝐹𝐴) ⊆ (𝐹𝐵))))
13 ssid 4017 . . . 4 (𝐹𝐵) ⊆ (𝐹𝐵)
14132a1i 12 . . 3 (𝐵 ∈ ω → (𝜑 → (𝐹𝐵) ⊆ (𝐹𝐵)))
15 isf32lem.b . . . . . . 7 (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))
16 suceq 6451 . . . . . . . . . 10 (𝑥 = 𝑏 → suc 𝑥 = suc 𝑏)
1716fveq2d 6910 . . . . . . . . 9 (𝑥 = 𝑏 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑏))
18 fveq2 6906 . . . . . . . . 9 (𝑥 = 𝑏 → (𝐹𝑥) = (𝐹𝑏))
1917, 18sseq12d 4028 . . . . . . . 8 (𝑥 = 𝑏 → ((𝐹‘suc 𝑥) ⊆ (𝐹𝑥) ↔ (𝐹‘suc 𝑏) ⊆ (𝐹𝑏)))
2019rspcv 3617 . . . . . . 7 (𝑏 ∈ ω → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥) → (𝐹‘suc 𝑏) ⊆ (𝐹𝑏)))
2115, 20syl5 34 . . . . . 6 (𝑏 ∈ ω → (𝜑 → (𝐹‘suc 𝑏) ⊆ (𝐹𝑏)))
2221ad2antrr 726 . . . . 5 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑏) → (𝜑 → (𝐹‘suc 𝑏) ⊆ (𝐹𝑏)))
23 sstr2 4001 . . . . 5 ((𝐹‘suc 𝑏) ⊆ (𝐹𝑏) → ((𝐹𝑏) ⊆ (𝐹𝐵) → (𝐹‘suc 𝑏) ⊆ (𝐹𝐵)))
2422, 23syl6 35 . . . 4 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑏) → (𝜑 → ((𝐹𝑏) ⊆ (𝐹𝐵) → (𝐹‘suc 𝑏) ⊆ (𝐹𝐵))))
2524a2d 29 . . 3 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑏) → ((𝜑 → (𝐹𝑏) ⊆ (𝐹𝐵)) → (𝜑 → (𝐹‘suc 𝑏) ⊆ (𝐹𝐵))))
263, 6, 9, 12, 14, 25findsg 7919 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → (𝜑 → (𝐹𝐴) ⊆ (𝐹𝐵)))
2726impr 454 1 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝐴𝜑)) → (𝐹𝐴) ⊆ (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1536  wcel 2105  wral 3058  wss 3962  𝒫 cpw 4604   cint 4950  ran crn 5689  suc csuc 6387  wf 6558  cfv 6562  ωcom 7886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fv 6570  df-om 7887
This theorem is referenced by:  isf32lem2  10391  isf32lem3  10392
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