Theorem hstrlem2 32188

Description: Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
hstrlem2.1	⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((projℎ‘𝑥)‘𝑢))

Assertion

Ref	Expression
hstrlem2	⊢ (𝐶 ∈ Cℋ → (𝑆‘𝐶) = ((projℎ‘𝐶)‘𝑢))

Distinct variable groups:   𝑥,𝐶   𝑥,𝑢

Allowed substitution hints:   𝐶(𝑢)   𝑆(𝑥,𝑢)

Proof of Theorem hstrlem2

Step	Hyp	Ref	Expression
1		fveq2 6858	. . 3 ⊢ (𝑥 = 𝐶 → (projℎ‘𝑥) = (projℎ‘𝐶))
2	1	fveq1d 6860	. 2 ⊢ (𝑥 = 𝐶 → ((projℎ‘𝑥)‘𝑢) = ((projℎ‘𝐶)‘𝑢))
3		hstrlem2.1	. 2 ⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((projℎ‘𝑥)‘𝑢))
4		fvex 6871	. 2 ⊢ ((projℎ‘𝐶)‘𝑢) ∈ V
5	2, 3, 4	fvmpt 6968	1 ⊢ (𝐶 ∈ Cℋ → (𝑆‘𝐶) = ((projℎ‘𝐶)‘𝑢))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ↦ cmpt 5188  ‘cfv 6511   C_ℋ cch 30858  proj_ℎcpjh 30866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519

This theorem is referenced by:  hstrlem3a  32189  hstrlem4  32191  hstrlem5  32192